L(s) = 1 | + 3-s − 2.74·5-s + 9-s − 1.54·11-s − 6.03·13-s − 2.74·15-s − 7.49·17-s − 6.03·19-s + 7.49·23-s + 2.54·25-s + 27-s + 1.25·29-s + 5.29·31-s − 1.54·33-s + 4.94·37-s − 6.03·39-s + 5.08·41-s − 3.45·43-s − 2.74·45-s + 9.49·47-s − 7.49·51-s − 3.83·53-s + 4.23·55-s − 6.03·57-s + 5.54·59-s + 14.5·61-s + 16.5·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.22·5-s + 0.333·9-s − 0.465·11-s − 1.67·13-s − 0.709·15-s − 1.81·17-s − 1.38·19-s + 1.56·23-s + 0.508·25-s + 0.192·27-s + 0.232·29-s + 0.950·31-s − 0.268·33-s + 0.813·37-s − 0.966·39-s + 0.794·41-s − 0.527·43-s − 0.409·45-s + 1.38·47-s − 1.04·51-s − 0.526·53-s + 0.571·55-s − 0.799·57-s + 0.721·59-s + 1.86·61-s + 2.05·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.094228099\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.094228099\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 2.74T + 5T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 + 7.49T + 17T^{2} \) |
| 19 | \( 1 + 6.03T + 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 - 1.25T + 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 - 4.94T + 37T^{2} \) |
| 41 | \( 1 - 5.08T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 - 9.49T + 47T^{2} \) |
| 53 | \( 1 + 3.83T + 53T^{2} \) |
| 59 | \( 1 - 5.54T + 59T^{2} \) |
| 61 | \( 1 - 14.5T + 61T^{2} \) |
| 67 | \( 1 - 4.03T + 67T^{2} \) |
| 71 | \( 1 - 5.49T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 7.79T + 79T^{2} \) |
| 83 | \( 1 + 6.52T + 83T^{2} \) |
| 89 | \( 1 - 9.49T + 89T^{2} \) |
| 97 | \( 1 + 1.54T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.366359861672425989603157539688, −7.54060323735456651837810979878, −7.07415820956782095813385149489, −6.37984307927286694809904602665, −5.00199051123414235311694269462, −4.52053108245893864886756175343, −3.89089382393204731167696561924, −2.68008481434923792509129963594, −2.32017412104279262723917433897, −0.52916244062833918146348448157,
0.52916244062833918146348448157, 2.32017412104279262723917433897, 2.68008481434923792509129963594, 3.89089382393204731167696561924, 4.52053108245893864886756175343, 5.00199051123414235311694269462, 6.37984307927286694809904602665, 7.07415820956782095813385149489, 7.54060323735456651837810979878, 8.366359861672425989603157539688