| L(s) = 1 | − 0.782·3-s + 2.74·5-s − 3.13·7-s − 2.38·9-s + 4.20·11-s + 4.79·13-s − 2.14·15-s − 4.78·17-s − 6.84·19-s + 2.45·21-s + 2.52·23-s + 2.51·25-s + 4.21·27-s − 0.824·29-s − 4.65·31-s − 3.29·33-s − 8.59·35-s − 5.55·37-s − 3.75·39-s + 6.84·41-s − 9.50·43-s − 6.54·45-s − 3.54·47-s + 2.82·49-s + 3.74·51-s + 7.03·53-s + 11.5·55-s + ⋯ |
| L(s) = 1 | − 0.451·3-s + 1.22·5-s − 1.18·7-s − 0.795·9-s + 1.26·11-s + 1.33·13-s − 0.553·15-s − 1.16·17-s − 1.57·19-s + 0.535·21-s + 0.525·23-s + 0.503·25-s + 0.811·27-s − 0.153·29-s − 0.836·31-s − 0.573·33-s − 1.45·35-s − 0.912·37-s − 0.600·39-s + 1.06·41-s − 1.44·43-s − 0.976·45-s − 0.517·47-s + 0.403·49-s + 0.524·51-s + 0.966·53-s + 1.55·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 0.782T + 3T^{2} \) |
| 5 | \( 1 - 2.74T + 5T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 - 4.79T + 13T^{2} \) |
| 17 | \( 1 + 4.78T + 17T^{2} \) |
| 19 | \( 1 + 6.84T + 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 + 0.824T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 + 5.55T + 37T^{2} \) |
| 41 | \( 1 - 6.84T + 41T^{2} \) |
| 43 | \( 1 + 9.50T + 43T^{2} \) |
| 47 | \( 1 + 3.54T + 47T^{2} \) |
| 53 | \( 1 - 7.03T + 53T^{2} \) |
| 59 | \( 1 + 0.838T + 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 - 3.63T + 67T^{2} \) |
| 71 | \( 1 + 7.42T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 + 2.08T + 79T^{2} \) |
| 83 | \( 1 + 2.93T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.421070483502966345077957777438, −6.81461689190000040981101539780, −6.48768049142050295114636626850, −6.07795221446708498101410708463, −5.34702589964333478357573446268, −4.18085527735457487541730563195, −3.43848264637217793464047402280, −2.39685859625477539538300930759, −1.45729627696579440546920030049, 0,
1.45729627696579440546920030049, 2.39685859625477539538300930759, 3.43848264637217793464047402280, 4.18085527735457487541730563195, 5.34702589964333478357573446268, 6.07795221446708498101410708463, 6.48768049142050295114636626850, 6.81461689190000040981101539780, 8.421070483502966345077957777438
