| L(s) = 1 | − 0.414·3-s + 5-s − 2.82·9-s + 11-s − 2.41·13-s − 0.414·15-s + 0.414·17-s + 2·19-s + 2.24·23-s + 25-s + 2.41·27-s + 29-s + 1.75·31-s − 0.414·33-s − 7.89·37-s + 0.999·39-s − 7.41·41-s + 0.343·43-s − 2.82·45-s + 10.4·47-s − 0.171·51-s + 9.41·53-s + 55-s − 0.828·57-s + 10.2·59-s + 1.17·61-s − 2.41·65-s + ⋯ |
| L(s) = 1 | − 0.239·3-s + 0.447·5-s − 0.942·9-s + 0.301·11-s − 0.669·13-s − 0.106·15-s + 0.100·17-s + 0.458·19-s + 0.467·23-s + 0.200·25-s + 0.464·27-s + 0.185·29-s + 0.315·31-s − 0.0721·33-s − 1.29·37-s + 0.160·39-s − 1.15·41-s + 0.0523·43-s − 0.421·45-s + 1.51·47-s − 0.0240·51-s + 1.29·53-s + 0.134·55-s − 0.109·57-s + 1.33·59-s + 0.150·61-s − 0.299·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.615340701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.615340701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 - 0.414T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 2.24T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + 7.89T + 37T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 - 0.343T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 1.17T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 5.17T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 - 0.0710T + 97T^{2} \) |
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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
−8.700256457131114110362495517554, −7.67093056227453065762255424778, −6.95058862883532081958357124370, −6.23470670692480588538784654503, −5.39536539034783086301336818665, −4.97038103860650556036123884087, −3.78756830819364628679704000857, −2.90923026907455797576820958185, −2.04870677627656162068820210702, −0.73080096487966109812042463586,
0.73080096487966109812042463586, 2.04870677627656162068820210702, 2.90923026907455797576820958185, 3.78756830819364628679704000857, 4.97038103860650556036123884087, 5.39536539034783086301336818665, 6.23470670692480588538784654503, 6.95058862883532081958357124370, 7.67093056227453065762255424778, 8.700256457131114110362495517554
