| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 4·11-s + 13-s + 14-s + 16-s − 2·17-s + 4·19-s − 20-s − 4·22-s + 25-s − 26-s − 28-s − 6·29-s − 8·31-s − 32-s + 2·34-s + 35-s + 6·37-s − 4·38-s + 40-s − 10·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 1/5·25-s − 0.196·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s + 0.342·34-s + 0.169·35-s + 0.986·37-s − 0.648·38-s + 0.158·40-s − 1.56·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 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
−7.36688409352237133317967282192, −7.02512941738273723966300003733, −6.20444860311355865353783501662, −5.59474973516163291410602015265, −4.54844690613169035912566655487, −3.71223813691433494629790156208, −3.19210289388837470837983732734, −2.00640492989058564959024791695, −1.17078312394837740567529576657, 0,
1.17078312394837740567529576657, 2.00640492989058564959024791695, 3.19210289388837470837983732734, 3.71223813691433494629790156208, 4.54844690613169035912566655487, 5.59474973516163291410602015265, 6.20444860311355865353783501662, 7.02512941738273723966300003733, 7.36688409352237133317967282192
