| L(s) = 1 | − 3·3-s + 7-s + 6·9-s + 5·11-s + 5·13-s + 7·17-s + 2·19-s − 3·21-s − 2·23-s − 9·27-s + 7·29-s − 4·31-s − 15·33-s + 6·37-s − 15·39-s − 12·41-s − 2·43-s + 47-s + 49-s − 21·51-s − 6·57-s + 4·59-s + 4·61-s + 6·63-s + 8·67-s + 6·69-s − 6·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.377·7-s + 2·9-s + 1.50·11-s + 1.38·13-s + 1.69·17-s + 0.458·19-s − 0.654·21-s − 0.417·23-s − 1.73·27-s + 1.29·29-s − 0.718·31-s − 2.61·33-s + 0.986·37-s − 2.40·39-s − 1.87·41-s − 0.304·43-s + 0.145·47-s + 1/7·49-s − 2.94·51-s − 0.794·57-s + 0.520·59-s + 0.512·61-s + 0.755·63-s + 0.977·67-s + 0.722·69-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.418753993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418753993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.795758213468663956094183111092, −7.984482054980256459193096634432, −6.97670663684812382164401826864, −6.39460459437087758891198231048, −5.76481244688009992103280329559, −5.12180714528748314961399662749, −4.15041618199622899610530288269, −3.45165323807794852534236235819, −1.47823230313481939170754494140, −0.949301332029487315300827868877,
0.949301332029487315300827868877, 1.47823230313481939170754494140, 3.45165323807794852534236235819, 4.15041618199622899610530288269, 5.12180714528748314961399662749, 5.76481244688009992103280329559, 6.39460459437087758891198231048, 6.97670663684812382164401826864, 7.984482054980256459193096634432, 8.795758213468663956094183111092
