L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s − 4·13-s − 14-s + 16-s + 8·19-s + 20-s − 22-s − 6·23-s + 25-s + 4·26-s + 28-s − 6·29-s − 4·31-s − 32-s + 35-s + 8·37-s − 8·38-s − 40-s − 6·41-s − 4·43-s + 44-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 + 0.301·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.83·19-s + 0.223·20-s − 0.213·22-s − 1.25·23-s + 1/5·25-s + 0.784·26-s + 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.169·35-s + 1.31·37-s − 1.29·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.81752417362046682051069698942, −7.03025030412999983562146699711, −6.33721947069175663086370447900, −5.45154142479466127801602110347, −4.98748404716723897288691138516, −3.86237526202351604499488780057, −2.99802519574418243402002440629, −2.06213709093989049011135441724, −1.34928275383485163357868884392, 0,
1.34928275383485163357868884392, 2.06213709093989049011135441724, 2.99802519574418243402002440629, 3.86237526202351604499488780057, 4.98748404716723897288691138516, 5.45154142479466127801602110347, 6.33721947069175663086370447900, 7.03025030412999983562146699711, 7.81752417362046682051069698942