L(s) = 1 | − 2.56·3-s − 0.561·5-s + 5.12·7-s + 3.56·9-s + 11-s + 3.12·13-s + 1.43·15-s + 2·17-s + 4·19-s − 13.1·21-s − 6.56·23-s − 4.68·25-s − 1.43·27-s + 3.12·29-s + 1.43·31-s − 2.56·33-s − 2.87·35-s − 3.43·37-s − 8·39-s + 7.12·41-s − 1.12·43-s − 2.00·45-s − 8·47-s + 19.2·49-s − 5.12·51-s − 4.24·53-s − 0.561·55-s + ⋯ |
L(s) = 1 | − 1.47·3-s − 0.251·5-s + 1.93·7-s + 1.18·9-s + 0.301·11-s + 0.866·13-s + 0.371·15-s + 0.485·17-s + 0.917·19-s − 2.86·21-s − 1.36·23-s − 0.936·25-s − 0.276·27-s + 0.579·29-s + 0.258·31-s − 0.445·33-s − 0.486·35-s − 0.565·37-s − 1.28·39-s + 1.11·41-s − 0.171·43-s − 0.298·45-s − 1.16·47-s + 2.74·49-s − 0.717·51-s − 0.583·53-s − 0.0757·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8785686878\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8785686878\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 0.561T + 5T^{2} \) |
| 7 | \( 1 - 5.12T + 7T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 - 3.12T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 + 1.12T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 7.12T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 + 9.68T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−12.17740460870029754106339197030, −11.63153923016304543014339444360, −11.08081938383617651858653402754, −10.06325863941388502371022050853, −8.417808339452679823957684602232, −7.54408672173888238600522045388, −6.09017099219840355473709429747, −5.22418980835345202428701573010, −4.17295278428351881056531905460, −1.38722634706423543959195079168,
1.38722634706423543959195079168, 4.17295278428351881056531905460, 5.22418980835345202428701573010, 6.09017099219840355473709429747, 7.54408672173888238600522045388, 8.417808339452679823957684602232, 10.06325863941388502371022050853, 11.08081938383617651858653402754, 11.63153923016304543014339444360, 12.17740460870029754106339197030