| L(s) = 1 | + 2-s − 4-s + 2·5-s + 7-s − 3·8-s + 2·10-s − 4·11-s − 2·13-s + 14-s − 16-s − 2·17-s − 4·19-s − 2·20-s − 4·22-s − 25-s − 2·26-s − 28-s − 29-s − 8·31-s + 5·32-s − 2·34-s + 2·35-s − 10·37-s − 4·38-s − 6·40-s + 6·41-s + 12·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s − 1.06·8-s + 0.632·10-s − 1.20·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.392·26-s − 0.188·28-s − 0.185·29-s − 1.43·31-s + 0.883·32-s − 0.342·34-s + 0.338·35-s − 1.64·37-s − 0.648·38-s − 0.948·40-s + 0.937·41-s + 1.82·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1827 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1827 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−19.09457364985794, −18.55205674281973, −17.79003902993652, −17.50877141393781, −16.76748215707568, −15.67693002798884, −15.23335436062556, −14.37321750368580, −13.92134120716589, −13.30969301681599, −12.60673543804569, −12.24360153444879, −10.86390088649027, −10.59211353740616, −9.436880328734883, −9.082806523501985, −8.078177291112253, −7.268774935367258, −6.092470603331158, −5.548265673919248, −4.837572681315123, −4.029061328137380, −2.802880002148125, −1.964099516657620, 0,
1.964099516657620, 2.802880002148125, 4.029061328137380, 4.837572681315123, 5.548265673919248, 6.092470603331158, 7.268774935367258, 8.078177291112253, 9.082806523501985, 9.436880328734883, 10.59211353740616, 10.86390088649027, 12.24360153444879, 12.60673543804569, 13.30969301681599, 13.92134120716589, 14.37321750368580, 15.23335436062556, 15.67693002798884, 16.76748215707568, 17.50877141393781, 17.79003902993652, 18.55205674281973, 19.09457364985794
