| L(s) = 1 | + 1.73·3-s − 3.61·5-s − 3.69·7-s + 0.0104·9-s − 3.30·11-s − 0.935·13-s − 6.27·15-s − 7.33·17-s + 1.39·19-s − 6.40·21-s − 1.11·23-s + 8.08·25-s − 5.18·27-s − 0.862·29-s − 9.74·31-s − 5.72·33-s + 13.3·35-s − 7.53·37-s − 1.62·39-s + 5.35·41-s + 4.27·43-s − 0.0378·45-s + 1.64·47-s + 6.62·49-s − 12.7·51-s − 11.2·53-s + 11.9·55-s + ⋯ |
| L(s) = 1 | + 1.00·3-s − 1.61·5-s − 1.39·7-s + 0.00349·9-s − 0.995·11-s − 0.259·13-s − 1.62·15-s − 1.77·17-s + 0.320·19-s − 1.39·21-s − 0.232·23-s + 1.61·25-s − 0.998·27-s − 0.160·29-s − 1.75·31-s − 0.997·33-s + 2.25·35-s − 1.23·37-s − 0.259·39-s + 0.836·41-s + 0.651·43-s − 0.00564·45-s + 0.240·47-s + 0.947·49-s − 1.78·51-s − 1.54·53-s + 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1208209941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1208209941\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 3.30T + 11T^{2} \) |
| 13 | \( 1 + 0.935T + 13T^{2} \) |
| 17 | \( 1 + 7.33T + 17T^{2} \) |
| 19 | \( 1 - 1.39T + 19T^{2} \) |
| 23 | \( 1 + 1.11T + 23T^{2} \) |
| 29 | \( 1 + 0.862T + 29T^{2} \) |
| 31 | \( 1 + 9.74T + 31T^{2} \) |
| 37 | \( 1 + 7.53T + 37T^{2} \) |
| 41 | \( 1 - 5.35T + 41T^{2} \) |
| 43 | \( 1 - 4.27T + 43T^{2} \) |
| 47 | \( 1 - 1.64T + 47T^{2} \) |
| 53 | \( 1 + 11.2T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 4.08T + 61T^{2} \) |
| 67 | \( 1 + 7.59T + 67T^{2} \) |
| 71 | \( 1 - 5.08T + 71T^{2} \) |
| 73 | \( 1 - 2.99T + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 4.02T + 89T^{2} \) |
| 97 | \( 1 - 9.60T + 97T^{2} \) |
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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
−7.72070909534204343308911843922, −7.40426924960163164459092198322, −6.68682759824346447947371214348, −5.81194634083951201832458447154, −4.81770349157130329748176521472, −4.01937251422687761603035835454, −3.42911414613999921091152762136, −2.89781037779581317345307172820, −2.10248905181079818047573861279, −0.14839370009879563239475604011,
0.14839370009879563239475604011, 2.10248905181079818047573861279, 2.89781037779581317345307172820, 3.42911414613999921091152762136, 4.01937251422687761603035835454, 4.81770349157130329748176521472, 5.81194634083951201832458447154, 6.68682759824346447947371214348, 7.40426924960163164459092198322, 7.72070909534204343308911843922
