| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 3·9-s − 2·10-s − 11-s − 2·13-s + 16-s − 2·17-s − 3·18-s − 2·20-s − 22-s − 8·23-s − 25-s − 2·26-s − 2·29-s + 8·31-s + 32-s − 2·34-s − 3·36-s − 2·37-s − 2·40-s − 10·41-s + 4·43-s − 44-s + 6·45-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 9-s − 0.632·10-s − 0.301·11-s − 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.447·20-s − 0.213·22-s − 1.66·23-s − 1/5·25-s − 0.392·26-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 0.328·37-s − 0.316·40-s − 1.56·41-s + 0.609·43-s − 0.150·44-s + 0.894·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.539853129925515428738601358433, −8.286432551019620936992312172883, −7.946251972761263727794474510252, −6.84495316961593084272070011502, −5.98669759336071997385266304726, −5.05189708017292657745820670233, −4.15526410391116979896636624709, −3.23491690494547198778904949011, −2.18198630913107193990538575660, 0,
2.18198630913107193990538575660, 3.23491690494547198778904949011, 4.15526410391116979896636624709, 5.05189708017292657745820670233, 5.98669759336071997385266304726, 6.84495316961593084272070011502, 7.946251972761263727794474510252, 8.286432551019620936992312172883, 9.539853129925515428738601358433
