| L(s) = 1 | + 1.79·3-s + 5-s − 1.07·7-s + 0.226·9-s − 2.15·11-s + 2.41·13-s + 1.79·15-s + 0.379·17-s − 1.31·19-s − 1.92·21-s − 2.25·23-s + 25-s − 4.98·27-s + 6.49·29-s − 8.43·31-s − 3.87·33-s − 1.07·35-s − 6.29·37-s + 4.33·39-s − 3.22·41-s + 9.67·43-s + 0.226·45-s − 13.3·47-s − 5.85·49-s + 0.681·51-s − 6.43·53-s − 2.15·55-s + ⋯ |
| L(s) = 1 | + 1.03·3-s + 0.447·5-s − 0.404·7-s + 0.0756·9-s − 0.650·11-s + 0.668·13-s + 0.463·15-s + 0.0920·17-s − 0.300·19-s − 0.419·21-s − 0.470·23-s + 0.200·25-s − 0.958·27-s + 1.20·29-s − 1.51·31-s − 0.674·33-s − 0.181·35-s − 1.03·37-s + 0.693·39-s − 0.504·41-s + 1.47·43-s + 0.0338·45-s − 1.95·47-s − 0.836·49-s + 0.0954·51-s − 0.883·53-s − 0.290·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 1.79T + 3T^{2} \) |
| 7 | \( 1 + 1.07T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 - 0.379T + 17T^{2} \) |
| 19 | \( 1 + 1.31T + 19T^{2} \) |
| 23 | \( 1 + 2.25T + 23T^{2} \) |
| 29 | \( 1 - 6.49T + 29T^{2} \) |
| 31 | \( 1 + 8.43T + 31T^{2} \) |
| 37 | \( 1 + 6.29T + 37T^{2} \) |
| 41 | \( 1 + 3.22T + 41T^{2} \) |
| 43 | \( 1 - 9.67T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 6.43T + 53T^{2} \) |
| 59 | \( 1 - 3.80T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 - 0.910T + 71T^{2} \) |
| 73 | \( 1 + 8.55T + 73T^{2} \) |
| 79 | \( 1 - 8.81T + 79T^{2} \) |
| 83 | \( 1 + 0.0616T + 83T^{2} \) |
| 89 | \( 1 - 3.54T + 89T^{2} \) |
| 97 | \( 1 - 2.28T + 97T^{2} \) |
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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
−7.70444493717736891596610202145, −6.79189066519379677110780591645, −6.14890235757879748161384980065, −5.44723796662629103362399987280, −4.62586804766069835474368501798, −3.59470932127955284723974616814, −3.14810936574956846355373474994, −2.30554499785473018521659627959, −1.54503314387467651978024330733, 0,
1.54503314387467651978024330733, 2.30554499785473018521659627959, 3.14810936574956846355373474994, 3.59470932127955284723974616814, 4.62586804766069835474368501798, 5.44723796662629103362399987280, 6.14890235757879748161384980065, 6.79189066519379677110780591645, 7.70444493717736891596610202145
