| L(s) = 1 | − 2.09·3-s − 2.18·5-s − 7-s + 1.40·9-s − 11-s + 13-s + 4.57·15-s − 0.842·17-s + 1.25·19-s + 2.09·21-s − 3.10·23-s − 0.240·25-s + 3.34·27-s + 2.98·29-s + 9.84·31-s + 2.09·33-s + 2.18·35-s + 4.05·37-s − 2.09·39-s − 2.29·41-s − 8.19·43-s − 3.06·45-s + 7.75·47-s + 49-s + 1.76·51-s + 7.37·53-s + 2.18·55-s + ⋯ |
| L(s) = 1 | − 1.21·3-s − 0.975·5-s − 0.377·7-s + 0.468·9-s − 0.301·11-s + 0.277·13-s + 1.18·15-s − 0.204·17-s + 0.287·19-s + 0.458·21-s − 0.648·23-s − 0.0480·25-s + 0.644·27-s + 0.553·29-s + 1.76·31-s + 0.365·33-s + 0.368·35-s + 0.666·37-s − 0.336·39-s − 0.358·41-s − 1.24·43-s − 0.457·45-s + 1.13·47-s + 0.142·49-s + 0.247·51-s + 1.01·53-s + 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.09T + 3T^{2} \) |
| 5 | \( 1 + 2.18T + 5T^{2} \) |
| 17 | \( 1 + 0.842T + 17T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 + 3.10T + 23T^{2} \) |
| 29 | \( 1 - 2.98T + 29T^{2} \) |
| 31 | \( 1 - 9.84T + 31T^{2} \) |
| 37 | \( 1 - 4.05T + 37T^{2} \) |
| 41 | \( 1 + 2.29T + 41T^{2} \) |
| 43 | \( 1 + 8.19T + 43T^{2} \) |
| 47 | \( 1 - 7.75T + 47T^{2} \) |
| 53 | \( 1 - 7.37T + 53T^{2} \) |
| 59 | \( 1 + 0.236T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 3.89T + 67T^{2} \) |
| 71 | \( 1 - 8.81T + 71T^{2} \) |
| 73 | \( 1 - 6.01T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 9.98T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.072895410815763816352445974066, −7.26888967192144043645473507122, −6.46252263370401911685318525076, −5.97762402014088127403425525236, −5.05844719639913031633849458079, −4.40291876957286648353284396408, −3.55667959644010485513682329647, −2.55876928705596003985203625979, −0.984990099649009665865325892026, 0,
0.984990099649009665865325892026, 2.55876928705596003985203625979, 3.55667959644010485513682329647, 4.40291876957286648353284396408, 5.05844719639913031633849458079, 5.97762402014088127403425525236, 6.46252263370401911685318525076, 7.26888967192144043645473507122, 8.072895410815763816352445974066
