| L(s) = 1 | + 2-s + 4-s + 2.82·5-s + 8-s + 2.82·10-s − 11-s + 5.65·13-s + 16-s − 2.82·17-s + 8.48·19-s + 2.82·20-s − 22-s + 8·23-s + 3.00·25-s + 5.65·26-s + 6·29-s − 8.48·31-s + 32-s − 2.82·34-s − 6·37-s + 8.48·38-s + 2.82·40-s + 8.48·41-s − 4·43-s − 44-s + 8·46-s + 2.82·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.26·5-s + 0.353·8-s + 0.894·10-s − 0.301·11-s + 1.56·13-s + 0.250·16-s − 0.685·17-s + 1.94·19-s + 0.632·20-s − 0.213·22-s + 1.66·23-s + 0.600·25-s + 1.10·26-s + 1.11·29-s − 1.52·31-s + 0.176·32-s − 0.485·34-s − 0.986·37-s + 1.37·38-s + 0.447·40-s + 1.32·41-s − 0.609·43-s − 0.150·44-s + 1.17·46-s + 0.412·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.298116160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.298116160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 8.48T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.43209077333041099967638462814, −6.84863709704885187339933982804, −6.18840944873326221476897663537, −5.49894036792827700017702860202, −5.20072340583974363770917662867, −4.23265279284282182639556182765, −3.28305089443051999549225027440, −2.80796549917341701180214480976, −1.71668467982582196674411124200, −1.08965411997081971373205071194,
1.08965411997081971373205071194, 1.71668467982582196674411124200, 2.80796549917341701180214480976, 3.28305089443051999549225027440, 4.23265279284282182639556182765, 5.20072340583974363770917662867, 5.49894036792827700017702860202, 6.18840944873326221476897663537, 6.84863709704885187339933982804, 7.43209077333041099967638462814
