| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 0.497·7-s − 2.23·8-s − 0.543·11-s − 0.335·13-s + 0.307·14-s + 1.85·16-s + 3.70·17-s − 5.04·19-s − 0.335·22-s + 2.75·23-s − 0.207·26-s − 0.804·28-s − 2.68·29-s + 1.31·31-s + 5.61·32-s + 2.28·34-s − 37-s − 3.11·38-s + 6.33·41-s − 0.971·43-s + 0.879·44-s + 1.70·46-s − 0.0176·47-s − 6.75·49-s + 0.543·52-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 0.187·7-s − 0.790·8-s − 0.163·11-s − 0.0931·13-s + 0.0821·14-s + 0.463·16-s + 0.898·17-s − 1.15·19-s − 0.0715·22-s + 0.575·23-s − 0.0407·26-s − 0.152·28-s − 0.498·29-s + 0.235·31-s + 0.993·32-s + 0.392·34-s − 0.164·37-s − 0.505·38-s + 0.989·41-s − 0.148·43-s + 0.132·44-s + 0.251·46-s − 0.00257·47-s − 0.964·49-s + 0.0753·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 7 | \( 1 - 0.497T + 7T^{2} \) |
| 11 | \( 1 + 0.543T + 11T^{2} \) |
| 13 | \( 1 + 0.335T + 13T^{2} \) |
| 17 | \( 1 - 3.70T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 - 2.75T + 23T^{2} \) |
| 29 | \( 1 + 2.68T + 29T^{2} \) |
| 31 | \( 1 - 1.31T + 31T^{2} \) |
| 41 | \( 1 - 6.33T + 41T^{2} \) |
| 43 | \( 1 + 0.971T + 43T^{2} \) |
| 47 | \( 1 + 0.0176T + 47T^{2} \) |
| 53 | \( 1 + 8.08T + 53T^{2} \) |
| 59 | \( 1 + 8.27T + 59T^{2} \) |
| 61 | \( 1 + 6.39T + 61T^{2} \) |
| 67 | \( 1 - 2.63T + 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 6.27T + 73T^{2} \) |
| 79 | \( 1 - 4.25T + 79T^{2} \) |
| 83 | \( 1 - 9.43T + 83T^{2} \) |
| 89 | \( 1 - 2.98T + 89T^{2} \) |
| 97 | \( 1 - 8.13T + 97T^{2} \) |
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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
−7.67338727274386824687402104311, −6.54316480026255905774689528122, −6.05402807965755861300128016113, −5.14771842807007870955417898493, −4.77755322227840417673507861874, −3.89884502789810023716696634245, −3.28868699208975564969141381173, −2.34971390056214445674533412531, −1.18087707528260209354115800105, 0,
1.18087707528260209354115800105, 2.34971390056214445674533412531, 3.28868699208975564969141381173, 3.89884502789810023716696634245, 4.77755322227840417673507861874, 5.14771842807007870955417898493, 6.05402807965755861300128016113, 6.54316480026255905774689528122, 7.67338727274386824687402104311
