| L(s) = 1 | + 4·3-s + 6·9-s + 16·17-s − 4·27-s + 24·47-s + 6·49-s + 64·51-s + 48·79-s − 37·81-s − 40·83-s − 8·109-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 96·141-s + 24·147-s + 149-s + 151-s + 96·153-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s + 3.88·17-s − 0.769·27-s + 3.50·47-s + 6/7·49-s + 8.96·51-s + 5.40·79-s − 4.11·81-s − 4.39·83-s − 0.766·109-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8.08·141-s + 1.97·147-s + 0.0819·149-s + 0.0813·151-s + 7.76·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.76·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.38653620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.38653620\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.52010210179974891981294856239, −6.10761152260331602636290958478, −5.99988915427465880613300077785, −5.83505520196398056942323751083, −5.55932277456644856315615244037, −5.50160908928694922955219877676, −5.29876926691340120909129357250, −5.03699694359572126519009859882, −4.81588145847436760520036011753, −4.18664215981407121322338641075, −4.15344920555672755489392341984, −4.10647763663342595868524236750, −3.85421372371903882723711967465, −3.38632192914247224779042283096, −3.26764716892190751106070420275, −3.23489457525188049906618901211, −2.98750901658983860713578081899, −2.69728818112346763175590312435, −2.36812941506272883274328947823, −2.17486628800189304740489406178, −1.97223543487132668589267210748, −1.48081326345212060102062957224, −1.25301348025842130937741264451, −0.790199277416321031072224109691, −0.59906208980625665939897346550,
0.59906208980625665939897346550, 0.790199277416321031072224109691, 1.25301348025842130937741264451, 1.48081326345212060102062957224, 1.97223543487132668589267210748, 2.17486628800189304740489406178, 2.36812941506272883274328947823, 2.69728818112346763175590312435, 2.98750901658983860713578081899, 3.23489457525188049906618901211, 3.26764716892190751106070420275, 3.38632192914247224779042283096, 3.85421372371903882723711967465, 4.10647763663342595868524236750, 4.15344920555672755489392341984, 4.18664215981407121322338641075, 4.81588145847436760520036011753, 5.03699694359572126519009859882, 5.29876926691340120909129357250, 5.50160908928694922955219877676, 5.55932277456644856315615244037, 5.83505520196398056942323751083, 5.99988915427465880613300077785, 6.10761152260331602636290958478, 6.52010210179974891981294856239
