| L(s) = 1 | − 2-s + 4-s + 3.70·5-s − 4.70·7-s − 8-s − 3.70·10-s − 4.70·11-s − 1.70·13-s + 4.70·14-s + 16-s + 2·17-s + 4·19-s + 3.70·20-s + 4.70·22-s + 8.70·25-s + 1.70·26-s − 4.70·28-s + 6.40·29-s + 0.701·31-s − 32-s − 2·34-s − 17.4·35-s + 3.40·37-s − 4·38-s − 3.70·40-s − 9.10·41-s + 1.40·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.65·5-s − 1.77·7-s − 0.353·8-s − 1.17·10-s − 1.41·11-s − 0.471·13-s + 1.25·14-s + 0.250·16-s + 0.485·17-s + 0.917·19-s + 0.827·20-s + 1.00·22-s + 1.74·25-s + 0.333·26-s − 0.888·28-s + 1.18·29-s + 0.126·31-s − 0.176·32-s − 0.342·34-s − 2.94·35-s + 0.559·37-s − 0.648·38-s − 0.585·40-s − 1.42·41-s + 0.213·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 3.70T + 5T^{2} \) |
| 7 | \( 1 + 4.70T + 7T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 + 1.70T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 6.40T + 29T^{2} \) |
| 31 | \( 1 - 0.701T + 31T^{2} \) |
| 37 | \( 1 - 3.40T + 37T^{2} \) |
| 41 | \( 1 + 9.10T + 41T^{2} \) |
| 43 | \( 1 - 1.40T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 6.40T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 2.59T + 71T^{2} \) |
| 73 | \( 1 + 6.40T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 4.29T + 89T^{2} \) |
| 97 | \( 1 - 7.80T + 97T^{2} \) |
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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
−7.31293532258043191028283361694, −6.56745584658287143430102394006, −6.16894216896839142690922871472, −5.45177567273947762763203757303, −4.90874949061948351406516943841, −3.36418017182379793474263492200, −2.83361468543916038633518761894, −2.28919686001279173821171776892, −1.14449191718667320904925705528, 0,
1.14449191718667320904925705528, 2.28919686001279173821171776892, 2.83361468543916038633518761894, 3.36418017182379793474263492200, 4.90874949061948351406516943841, 5.45177567273947762763203757303, 6.16894216896839142690922871472, 6.56745584658287143430102394006, 7.31293532258043191028283361694
