| L(s) = 1 | − 1.60·3-s + 2.47·5-s − 0.437·9-s + 1.48·11-s − 2.55·13-s − 3.95·15-s − 4.00·17-s − 1.95·19-s + 3.60·23-s + 1.11·25-s + 5.50·27-s + 3.95·29-s − 4.47·31-s − 2.37·33-s + 8.98·37-s + 4.08·39-s − 41-s − 2.00·43-s − 1.08·45-s + 0.268·47-s + 6.41·51-s + 2.47·53-s + 3.67·55-s + 3.12·57-s − 2.37·59-s − 3.24·61-s − 6.31·65-s + ⋯ |
| L(s) = 1 | − 0.924·3-s + 1.10·5-s − 0.145·9-s + 0.447·11-s − 0.708·13-s − 1.02·15-s − 0.971·17-s − 0.447·19-s + 0.750·23-s + 0.223·25-s + 1.05·27-s + 0.735·29-s − 0.803·31-s − 0.413·33-s + 1.47·37-s + 0.654·39-s − 0.156·41-s − 0.306·43-s − 0.161·45-s + 0.0391·47-s + 0.898·51-s + 0.339·53-s + 0.495·55-s + 0.413·57-s − 0.309·59-s − 0.416·61-s − 0.783·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 1.60T + 3T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 11 | \( 1 - 1.48T + 11T^{2} \) |
| 13 | \( 1 + 2.55T + 13T^{2} \) |
| 17 | \( 1 + 4.00T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 - 3.95T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 - 0.268T + 47T^{2} \) |
| 53 | \( 1 - 2.47T + 53T^{2} \) |
| 59 | \( 1 + 2.37T + 59T^{2} \) |
| 61 | \( 1 + 3.24T + 61T^{2} \) |
| 67 | \( 1 - 2.16T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 5.46T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 3.21T + 89T^{2} \) |
| 97 | \( 1 - 2.80T + 97T^{2} \) |
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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
−7.27396344584285602861072687064, −6.49709803036362630220307837429, −6.21546865101805650102383792514, −5.41856412904030942233651302487, −4.87354123093327141480020042413, −4.13109421683395964480608728833, −2.88231279182508606252620480657, −2.21739579887326355486513928343, −1.20529387162219020389485705432, 0,
1.20529387162219020389485705432, 2.21739579887326355486513928343, 2.88231279182508606252620480657, 4.13109421683395964480608728833, 4.87354123093327141480020042413, 5.41856412904030942233651302487, 6.21546865101805650102383792514, 6.49709803036362630220307837429, 7.27396344584285602861072687064
