| L(s) = 1 | + 0.790·2-s + 2.57·3-s − 1.37·4-s + 2.03·6-s + 0.0669·7-s − 2.66·8-s + 3.63·9-s − 11-s − 3.54·12-s + 4.14·13-s + 0.0528·14-s + 0.643·16-s − 6.61·17-s + 2.87·18-s − 19-s + 0.172·21-s − 0.790·22-s + 2.21·23-s − 6.86·24-s + 3.27·26-s + 1.62·27-s − 0.0920·28-s + 2.56·29-s + 8.77·31-s + 5.84·32-s − 2.57·33-s − 5.22·34-s + ⋯ |
| L(s) = 1 | + 0.558·2-s + 1.48·3-s − 0.687·4-s + 0.830·6-s + 0.0252·7-s − 0.943·8-s + 1.21·9-s − 0.301·11-s − 1.02·12-s + 1.15·13-s + 0.0141·14-s + 0.160·16-s − 1.60·17-s + 0.676·18-s − 0.229·19-s + 0.0376·21-s − 0.168·22-s + 0.461·23-s − 1.40·24-s + 0.642·26-s + 0.313·27-s − 0.0173·28-s + 0.476·29-s + 1.57·31-s + 1.03·32-s − 0.448·33-s − 0.896·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.676561179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.676561179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 0.790T + 2T^{2} \) |
| 3 | \( 1 - 2.57T + 3T^{2} \) |
| 7 | \( 1 - 0.0669T + 7T^{2} \) |
| 13 | \( 1 - 4.14T + 13T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 23 | \( 1 - 2.21T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 - 8.77T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 3.95T + 43T^{2} \) |
| 47 | \( 1 + 8.97T + 47T^{2} \) |
| 53 | \( 1 + 3.13T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 4.11T + 71T^{2} \) |
| 73 | \( 1 - 7.16T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 3.78T + 83T^{2} \) |
| 89 | \( 1 - 0.223T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.256357065043819532824637794429, −7.86995100060865302508956906694, −6.60799524956714545035590704346, −6.16746879655788994499661031841, −5.00478623129678181964985040318, −4.29983940188975405702386046857, −3.79502058605809742111776732880, −2.86955103111464929344163798260, −2.33313935757405480113689386897, −0.905530542850433538048393174798,
0.905530542850433538048393174798, 2.33313935757405480113689386897, 2.86955103111464929344163798260, 3.79502058605809742111776732880, 4.29983940188975405702386046857, 5.00478623129678181964985040318, 6.16746879655788994499661031841, 6.60799524956714545035590704346, 7.86995100060865302508956906694, 8.256357065043819532824637794429
