| L(s) = 1 | + 2·9-s − 12·17-s + 12·41-s − 28·49-s + 4·73-s + 9·81-s + 36·89-s − 40·97-s − 36·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 2.91·17-s + 1.87·41-s − 4·49-s + 0.468·73-s + 81-s + 3.81·89-s − 4.06·97-s − 3.38·113-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8051788419\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8051788419\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.12260008916806473155457670815, −6.11632274523036361135323372487, −5.76718477696392616676563607441, −5.41494446711332131873101037204, −5.34781597212091731735130827024, −4.99921577116159331332355433200, −4.95762288531116673030986609288, −4.69243630231824038262369112038, −4.52728873781391495617902336603, −4.16670901801288976727145591440, −4.13419129508342722091459242759, −4.04611958600484910536531856053, −3.66421006401019417141165307053, −3.47641566163354016037920353399, −2.96168533417526199989981647943, −2.91413828786457210999861216894, −2.89140266603604311305672398330, −2.23603266924591935745587948256, −2.15792020858860269943023734412, −2.08213856787426906996728661832, −1.67845939906725181624099824744, −1.32375590742799428679056681339, −1.13753538435333575524224412542, −0.57128140795724239532338152535, −0.16168061828592726358013354107,
0.16168061828592726358013354107, 0.57128140795724239532338152535, 1.13753538435333575524224412542, 1.32375590742799428679056681339, 1.67845939906725181624099824744, 2.08213856787426906996728661832, 2.15792020858860269943023734412, 2.23603266924591935745587948256, 2.89140266603604311305672398330, 2.91413828786457210999861216894, 2.96168533417526199989981647943, 3.47641566163354016037920353399, 3.66421006401019417141165307053, 4.04611958600484910536531856053, 4.13419129508342722091459242759, 4.16670901801288976727145591440, 4.52728873781391495617902336603, 4.69243630231824038262369112038, 4.95762288531116673030986609288, 4.99921577116159331332355433200, 5.34781597212091731735130827024, 5.41494446711332131873101037204, 5.76718477696392616676563607441, 6.11632274523036361135323372487, 6.12260008916806473155457670815
