| L(s) = 1 | + 43.6·3-s − 1.11e3·5-s − 2.40e3·7-s − 1.77e4·9-s + 8.94e4·11-s − 2.85e4·13-s − 4.88e4·15-s + 6.92e3·17-s + 9.53e5·19-s − 1.04e5·21-s − 2.19e5·23-s − 7.01e5·25-s − 1.63e6·27-s + 3.05e6·29-s + 7.36e6·31-s + 3.90e6·33-s + 2.68e6·35-s + 2.13e7·37-s − 1.24e6·39-s − 2.82e7·41-s + 3.92e7·43-s + 1.98e7·45-s + 6.24e7·47-s − 3.45e7·49-s + 3.02e5·51-s + 2.98e7·53-s − 1.00e8·55-s + ⋯ |
| L(s) = 1 | + 0.311·3-s − 0.800·5-s − 0.377·7-s − 0.903·9-s + 1.84·11-s − 0.277·13-s − 0.249·15-s + 0.0201·17-s + 1.67·19-s − 0.117·21-s − 0.163·23-s − 0.359·25-s − 0.592·27-s + 0.802·29-s + 1.43·31-s + 0.573·33-s + 0.302·35-s + 1.87·37-s − 0.0863·39-s − 1.55·41-s + 1.75·43-s + 0.723·45-s + 1.86·47-s − 0.857·49-s + 0.00625·51-s + 0.518·53-s − 1.47·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.806744608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.806744608\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 2.85e4T \) |
| good | 3 | \( 1 - 43.6T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.11e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 2.40e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.94e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 6.92e3T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.53e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.19e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.05e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.36e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.13e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.82e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.92e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.24e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.98e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.75e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 6.17e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.57e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.59e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.29e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.23e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.81e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.00e8T + 7.60e17T^{2} \) |
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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
−13.74930981573966667589958849106, −11.97093541535978727954117574960, −11.59624657489365734096518653663, −9.755401311924265746386569589838, −8.715006145296885801659986421143, −7.43769235501524633905343579264, −6.04991810302448144253546824460, −4.18012431776579134468606081144, −2.97308341239007804086345646705, −0.877728196482817077474045738327,
0.877728196482817077474045738327, 2.97308341239007804086345646705, 4.18012431776579134468606081144, 6.04991810302448144253546824460, 7.43769235501524633905343579264, 8.715006145296885801659986421143, 9.755401311924265746386569589838, 11.59624657489365734096518653663, 11.97093541535978727954117574960, 13.74930981573966667589958849106
