| L(s) = 1 | − 2.41·2-s + 3.82·4-s − 0.765·5-s + 2.61·7-s − 4.41·8-s + 1.84·10-s + 2.61·11-s + 1.41·13-s − 6.30·14-s + 2.99·16-s − 0.828·19-s − 2.93·20-s − 6.30·22-s − 4.77·23-s − 4.41·25-s − 3.41·26-s + 10.0·28-s + 0.317·29-s − 7.83·31-s + 1.58·32-s − 2·35-s − 9.23·37-s + 1.99·38-s + 3.37·40-s + 1.21·41-s + 0.828·43-s + 10.0·44-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s − 0.342·5-s + 0.987·7-s − 1.56·8-s + 0.584·10-s + 0.787·11-s + 0.392·13-s − 1.68·14-s + 0.749·16-s − 0.190·19-s − 0.655·20-s − 1.34·22-s − 0.996·23-s − 0.882·25-s − 0.669·26-s + 1.89·28-s + 0.0588·29-s − 1.40·31-s + 0.280·32-s − 0.338·35-s − 1.51·37-s + 0.324·38-s + 0.534·40-s + 0.189·41-s + 0.126·43-s + 1.50·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 0.765T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 + 4.77T + 23T^{2} \) |
| 29 | \( 1 - 0.317T + 29T^{2} \) |
| 31 | \( 1 + 7.83T + 31T^{2} \) |
| 37 | \( 1 + 9.23T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 - 0.828T + 43T^{2} \) |
| 47 | \( 1 + 5.17T + 47T^{2} \) |
| 53 | \( 1 + 1.41T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 3.82T + 61T^{2} \) |
| 67 | \( 1 + 1.17T + 67T^{2} \) |
| 71 | \( 1 + 5.41T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 4.77T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 6.58T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−8.446434075935156701390761504345, −7.987023465514528263769755137634, −7.29724326928955753197032993887, −6.53069359464380456716911808514, −5.62516016145817110146814546022, −4.43435537830566864855172263218, −3.49503014020186089314346317875, −2.01239700446173852361709789931, −1.44642306109216458325530240571, 0,
1.44642306109216458325530240571, 2.01239700446173852361709789931, 3.49503014020186089314346317875, 4.43435537830566864855172263218, 5.62516016145817110146814546022, 6.53069359464380456716911808514, 7.29724326928955753197032993887, 7.987023465514528263769755137634, 8.446434075935156701390761504345
