| L(s) = 1 | − 2·4-s + 5-s + 2·7-s − 3·11-s + 2·13-s + 4·16-s + 2·19-s − 2·20-s − 3·23-s + 25-s − 4·28-s + 29-s + 8·31-s + 2·35-s − 37-s + 3·41-s − 43-s + 6·44-s + 6·47-s − 3·49-s − 4·52-s + 3·53-s − 3·55-s + 12·59-s + 8·61-s − 8·64-s + 2·65-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.755·7-s − 0.904·11-s + 0.554·13-s + 16-s + 0.458·19-s − 0.447·20-s − 0.625·23-s + 1/5·25-s − 0.755·28-s + 0.185·29-s + 1.43·31-s + 0.338·35-s − 0.164·37-s + 0.468·41-s − 0.152·43-s + 0.904·44-s + 0.875·47-s − 3/7·49-s − 0.554·52-s + 0.412·53-s − 0.404·55-s + 1.56·59-s + 1.02·61-s − 64-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.500117804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500117804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−9.788906746241101084288007436947, −8.642814857707797957392115254585, −8.272248506160292840175320854368, −7.38238509669578907211253317909, −6.12295429159013140281371744854, −5.32855597269579923033160745483, −4.66262168952595729437838996730, −3.65107489984939242445264232066, −2.36748279843542963780814696470, −0.942520552818180520165204014558,
0.942520552818180520165204014558, 2.36748279843542963780814696470, 3.65107489984939242445264232066, 4.66262168952595729437838996730, 5.32855597269579923033160745483, 6.12295429159013140281371744854, 7.38238509669578907211253317909, 8.272248506160292840175320854368, 8.642814857707797957392115254585, 9.788906746241101084288007436947
