| L(s) = 1 | + 2-s − 1.23·3-s + 4-s + 3.67·5-s − 1.23·6-s − 4.90·7-s + 8-s − 1.48·9-s + 3.67·10-s + 11-s − 1.23·12-s + 5.58·13-s − 4.90·14-s − 4.52·15-s + 16-s + 2.65·17-s − 1.48·18-s + 2.61·19-s + 3.67·20-s + 6.03·21-s + 22-s + 8.29·23-s − 1.23·24-s + 8.52·25-s + 5.58·26-s + 5.52·27-s − 4.90·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.710·3-s + 0.5·4-s + 1.64·5-s − 0.502·6-s − 1.85·7-s + 0.353·8-s − 0.494·9-s + 1.16·10-s + 0.301·11-s − 0.355·12-s + 1.54·13-s − 1.30·14-s − 1.16·15-s + 0.250·16-s + 0.642·17-s − 0.349·18-s + 0.598·19-s + 0.822·20-s + 1.31·21-s + 0.213·22-s + 1.72·23-s − 0.251·24-s + 1.70·25-s + 1.09·26-s + 1.06·27-s − 0.926·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 946 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.273148695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.273148695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 13 | \( 1 - 5.58T + 13T^{2} \) |
| 17 | \( 1 - 2.65T + 17T^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 - 8.29T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 + 4.98T + 37T^{2} \) |
| 41 | \( 1 + 4.31T + 41T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 3.39T + 53T^{2} \) |
| 59 | \( 1 - 4.30T + 59T^{2} \) |
| 61 | \( 1 - 4.48T + 61T^{2} \) |
| 67 | \( 1 - 0.438T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 1.10T + 73T^{2} \) |
| 79 | \( 1 + 0.722T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−10.11479076807835624774296477156, −9.408472592368287734864165601617, −8.632142876153011991232813070577, −6.70184253432444241409498187354, −6.56519276389232566312103191802, −5.70103420831367952390853491543, −5.22545685516091605525082662322, −3.48456072547737694590400182224, −2.87530836177853761768706651691, −1.21205831950267377008217253969,
1.21205831950267377008217253969, 2.87530836177853761768706651691, 3.48456072547737694590400182224, 5.22545685516091605525082662322, 5.70103420831367952390853491543, 6.56519276389232566312103191802, 6.70184253432444241409498187354, 8.632142876153011991232813070577, 9.408472592368287734864165601617, 10.11479076807835624774296477156
