| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 4·11-s + 12-s + 2·13-s + 14-s + 16-s − 17-s − 18-s + 4·19-s − 21-s − 4·22-s − 8·23-s − 24-s − 2·26-s + 27-s − 28-s + 6·29-s − 32-s + 4·33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.218·21-s − 0.852·22-s − 1.66·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.176·32-s + 0.696·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.245262112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.245262112\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.97627762879888, −15.50219286333085, −14.53216133217428, −14.30738376530713, −13.74102724854637, −13.12200009405910, −12.23540632788233, −12.07388506783804, −11.26835420738073, −10.75039432514925, −10.00172601099499, −9.441430792486892, −9.239366207568629, −8.372082169457220, −8.008571953145197, −7.316299599235274, −6.562697157369338, −6.224470952907654, −5.441842893900630, −4.299139086716852, −3.881646563946145, −3.081787593497868, −2.343777238769777, −1.477126050376943, −0.7283640792662358,
0.7283640792662358, 1.477126050376943, 2.343777238769777, 3.081787593497868, 3.881646563946145, 4.299139086716852, 5.441842893900630, 6.224470952907654, 6.562697157369338, 7.316299599235274, 8.008571953145197, 8.372082169457220, 9.239366207568629, 9.441430792486892, 10.00172601099499, 10.75039432514925, 11.26835420738073, 12.07388506783804, 12.23540632788233, 13.12200009405910, 13.74102724854637, 14.30738376530713, 14.53216133217428, 15.50219286333085, 15.97627762879888
