L(s) = 1 | + 0.844·3-s − 2.47·5-s − 0.972·7-s − 2.28·9-s + 3.50·11-s + 2.87·13-s − 2.08·15-s − 2.27·17-s − 6.78·19-s − 0.821·21-s + 3.58·23-s + 1.10·25-s − 4.46·27-s − 7.60·29-s − 0.279·31-s + 2.96·33-s + 2.40·35-s − 9.37·37-s + 2.43·39-s − 10.4·41-s − 5.70·43-s + 5.64·45-s − 8.12·47-s − 6.05·49-s − 1.91·51-s + 1.01·53-s − 8.66·55-s + ⋯ |
L(s) = 1 | + 0.487·3-s − 1.10·5-s − 0.367·7-s − 0.762·9-s + 1.05·11-s + 0.798·13-s − 0.538·15-s − 0.550·17-s − 1.55·19-s − 0.179·21-s + 0.746·23-s + 0.220·25-s − 0.859·27-s − 1.41·29-s − 0.0502·31-s + 0.516·33-s + 0.405·35-s − 1.54·37-s + 0.389·39-s − 1.63·41-s − 0.869·43-s + 0.841·45-s − 1.18·47-s − 0.864·49-s − 0.268·51-s + 0.139·53-s − 1.16·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 - 0.844T + 3T^{2} \) |
| 5 | \( 1 + 2.47T + 5T^{2} \) |
| 7 | \( 1 + 0.972T + 7T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 13 | \( 1 - 2.87T + 13T^{2} \) |
| 17 | \( 1 + 2.27T + 17T^{2} \) |
| 19 | \( 1 + 6.78T + 19T^{2} \) |
| 23 | \( 1 - 3.58T + 23T^{2} \) |
| 29 | \( 1 + 7.60T + 29T^{2} \) |
| 31 | \( 1 + 0.279T + 31T^{2} \) |
| 37 | \( 1 + 9.37T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 8.12T + 47T^{2} \) |
| 53 | \( 1 - 1.01T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 9.95T + 61T^{2} \) |
| 67 | \( 1 + 4.50T + 67T^{2} \) |
| 71 | \( 1 - 9.82T + 71T^{2} \) |
| 73 | \( 1 - 6.37T + 73T^{2} \) |
| 79 | \( 1 - 1.13T + 79T^{2} \) |
| 83 | \( 1 + 1.53T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 0.254T + 97T^{2} \) |
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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
−9.284995420653909621990144789855, −8.600828013499118515530172032613, −8.190519423354757049192539767356, −6.93535565994031682160009390707, −6.38802570344339383038221391291, −5.09308310374110988232426470136, −3.77393946183372340212793282603, −3.51598531164343996720076486474, −1.94711385585014768142236227981, 0,
1.94711385585014768142236227981, 3.51598531164343996720076486474, 3.77393946183372340212793282603, 5.09308310374110988232426470136, 6.38802570344339383038221391291, 6.93535565994031682160009390707, 8.190519423354757049192539767356, 8.600828013499118515530172032613, 9.284995420653909621990144789855