| L(s) = 1 | + 2.85·3-s + 1.91·5-s − 1.96·7-s + 5.15·9-s + 4.57·11-s + 3.48·13-s + 5.47·15-s + 3.56·17-s + 1.12·19-s − 5.62·21-s + 5.46·23-s − 1.32·25-s + 6.15·27-s − 3.43·29-s − 2.84·31-s + 13.0·33-s − 3.77·35-s + 9.69·37-s + 9.96·39-s + 8.38·41-s − 9.34·43-s + 9.87·45-s − 8.33·47-s − 3.12·49-s + 10.1·51-s − 14.0·53-s + 8.76·55-s + ⋯ |
| L(s) = 1 | + 1.64·3-s + 0.856·5-s − 0.744·7-s + 1.71·9-s + 1.37·11-s + 0.967·13-s + 1.41·15-s + 0.864·17-s + 0.258·19-s − 1.22·21-s + 1.13·23-s − 0.265·25-s + 1.18·27-s − 0.637·29-s − 0.511·31-s + 2.27·33-s − 0.637·35-s + 1.59·37-s + 1.59·39-s + 1.30·41-s − 1.42·43-s + 1.47·45-s − 1.21·47-s − 0.446·49-s + 1.42·51-s − 1.93·53-s + 1.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.343922458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.343922458\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.85T + 3T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 31 | \( 1 + 2.84T + 31T^{2} \) |
| 37 | \( 1 - 9.69T + 37T^{2} \) |
| 41 | \( 1 - 8.38T + 41T^{2} \) |
| 43 | \( 1 + 9.34T + 43T^{2} \) |
| 47 | \( 1 + 8.33T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 + 6.01T + 67T^{2} \) |
| 71 | \( 1 - 0.803T + 71T^{2} \) |
| 73 | \( 1 - 4.79T + 73T^{2} \) |
| 79 | \( 1 + 4.21T + 79T^{2} \) |
| 83 | \( 1 + 2.93T + 83T^{2} \) |
| 89 | \( 1 - 9.76T + 89T^{2} \) |
| 97 | \( 1 + 0.568T + 97T^{2} \) |
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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
−7.890284416836642052734069428557, −7.27231531998516275557296374576, −6.36826613310439979469916371185, −6.06652517334642122786680833854, −4.92745227703919389447036588006, −3.88820592489016552297219805569, −3.42846988338910579370422598683, −2.82928861236925199255480770642, −1.78014002717213838727232191320, −1.19784640824329971369686054553,
1.19784640824329971369686054553, 1.78014002717213838727232191320, 2.82928861236925199255480770642, 3.42846988338910579370422598683, 3.88820592489016552297219805569, 4.92745227703919389447036588006, 6.06652517334642122786680833854, 6.36826613310439979469916371185, 7.27231531998516275557296374576, 7.890284416836642052734069428557
