| L(s) = 1 | + 2·2-s + 3·4-s + 8·8-s + 2·9-s + 9·16-s − 16·17-s + 4·18-s + 6·32-s − 32·34-s + 6·36-s − 36·49-s − 32·53-s − 8·61-s + 11·64-s − 48·68-s + 16·72-s + 2·81-s − 72·98-s − 64·106-s + 40·109-s + 112·113-s − 48·121-s − 16·122-s + 127-s − 12·128-s + 131-s − 128·136-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 2.82·8-s + 2/3·9-s + 9/4·16-s − 3.88·17-s + 0.942·18-s + 1.06·32-s − 5.48·34-s + 36-s − 5.14·49-s − 4.39·53-s − 1.02·61-s + 11/8·64-s − 5.82·68-s + 1.88·72-s + 2/9·81-s − 7.27·98-s − 6.21·106-s + 3.83·109-s + 10.5·113-s − 4.36·121-s − 1.44·122-s + 0.0887·127-s − 1.06·128-s + 0.0873·131-s − 10.9·136-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.227255684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.227255684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T - p T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 - 2 T^{2} + 2 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 + 18 T^{2} + 162 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 24 T^{2} + 318 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 16 T^{2} + 334 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 19 | \( ( 1 - 56 T^{2} + 1438 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 80 T^{2} + 3214 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 96 T^{2} + 4158 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 112 T^{2} + 5806 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 112 T^{2} + 5886 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 110 T^{2} + 5890 T^{4} + 110 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 178 T^{2} + 12322 T^{4} - 178 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 16 T^{2} + 1518 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 2 T + 106 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 222 T^{2} + 21282 T^{4} + 222 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 228 T^{2} + 22806 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 168 T^{2} + 19470 T^{4} - 168 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 82 T^{2} + 4834 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 212 T^{2} + 25990 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.29781060726282089725950437005, −4.85384875188578496568772188824, −4.74406889480368944187998791227, −4.70112310348902705814701297858, −4.67504970126721101903202572465, −4.62041650311635729780935191155, −4.53197446742712637270205907944, −4.32097337955188654482571814383, −4.21792540070786031498926910673, −4.00486847023652414018317593202, −3.75602648697378891801418330507, −3.38033596623352599878866711225, −3.33195707669791873931256695076, −3.19765502750783427585763843098, −3.15858736582851018107682320876, −3.14658849291479003509065990565, −2.45420318613351866694000462556, −2.39849004701654073599190469424, −2.10971197066226745619528438226, −1.97834830941150648391040170243, −1.91135949239542909448822821589, −1.79545870087808403391718245635, −1.35654984211100453819360860612, −1.16597243056715188675866212877, −0.17793161062741461146819179589,
0.17793161062741461146819179589, 1.16597243056715188675866212877, 1.35654984211100453819360860612, 1.79545870087808403391718245635, 1.91135949239542909448822821589, 1.97834830941150648391040170243, 2.10971197066226745619528438226, 2.39849004701654073599190469424, 2.45420318613351866694000462556, 3.14658849291479003509065990565, 3.15858736582851018107682320876, 3.19765502750783427585763843098, 3.33195707669791873931256695076, 3.38033596623352599878866711225, 3.75602648697378891801418330507, 4.00486847023652414018317593202, 4.21792540070786031498926910673, 4.32097337955188654482571814383, 4.53197446742712637270205907944, 4.62041650311635729780935191155, 4.67504970126721101903202572465, 4.70112310348902705814701297858, 4.74406889480368944187998791227, 4.85384875188578496568772188824, 5.29781060726282089725950437005
Plot not available for L-functions of degree greater than 10.
