| L(s) = 1 | + 0.747·2-s − 1.44·4-s − 3.07·7-s − 2.57·8-s − 2.70·11-s + 4.25·13-s − 2.29·14-s + 0.958·16-s − 3.18·17-s − 4.77·19-s − 2.02·22-s − 2.20·23-s + 3.18·26-s + 4.43·28-s − 9.00·29-s + 8.95·31-s + 5.86·32-s − 2.37·34-s − 37-s − 3.56·38-s + 5.38·41-s − 6.38·43-s + 3.90·44-s − 1.64·46-s − 8.04·47-s + 2.46·49-s − 6.13·52-s + ⋯ |
| L(s) = 1 | + 0.528·2-s − 0.720·4-s − 1.16·7-s − 0.909·8-s − 0.816·11-s + 1.18·13-s − 0.614·14-s + 0.239·16-s − 0.771·17-s − 1.09·19-s − 0.431·22-s − 0.459·23-s + 0.624·26-s + 0.837·28-s − 1.67·29-s + 1.60·31-s + 1.03·32-s − 0.407·34-s − 0.164·37-s − 0.578·38-s + 0.840·41-s − 0.974·43-s + 0.588·44-s − 0.242·46-s − 1.17·47-s + 0.351·49-s − 0.851·52-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
| \(L(1)\) |
\(\approx\) |
\(0.7383243709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7383243709\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.747T + 2T^{2} \) |
| 7 | \( 1 + 3.07T + 7T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 13 | \( 1 - 4.25T + 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 + 4.77T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 + 9.00T + 29T^{2} \) |
| 31 | \( 1 - 8.95T + 31T^{2} \) |
| 41 | \( 1 - 5.38T + 41T^{2} \) |
| 43 | \( 1 + 6.38T + 43T^{2} \) |
| 47 | \( 1 + 8.04T + 47T^{2} \) |
| 53 | \( 1 + 11.0T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 + 15.0T + 61T^{2} \) |
| 67 | \( 1 - 8.51T + 67T^{2} \) |
| 71 | \( 1 + 5.25T + 71T^{2} \) |
| 73 | \( 1 - 8.04T + 73T^{2} \) |
| 79 | \( 1 + 2.93T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 - 9.91T + 89T^{2} \) |
| 97 | \( 1 + 7.58T + 97T^{2} \) |
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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
−8.083328526473261724386490350909, −6.85818879966292268536642704156, −6.17945511115319863034516398423, −5.92006209004659999425576280075, −4.86664085318960686016158666003, −4.29920889841438084687541123763, −3.49113604653194727887005680480, −2.98970379595825070338671478254, −1.87970667724303601669316945631, −0.37296951214259763665603860381,
0.37296951214259763665603860381, 1.87970667724303601669316945631, 2.98970379595825070338671478254, 3.49113604653194727887005680480, 4.29920889841438084687541123763, 4.86664085318960686016158666003, 5.92006209004659999425576280075, 6.17945511115319863034516398423, 6.85818879966292268536642704156, 8.083328526473261724386490350909
