L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 8-s + 9-s + 2·10-s − 12-s − 2·13-s + 2·15-s + 16-s − 2·17-s − 18-s + 19-s − 2·20-s + 8·23-s + 24-s − 25-s + 2·26-s − 27-s + 2·29-s − 2·30-s − 4·31-s − 32-s + 2·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s − 0.447·20-s + 1.66·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.365·30-s − 0.718·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−7.75433809423225303178239940813, −7.01812692974366469345650590427, −6.75904741051958840400035550399, −5.58344756264802333727634053775, −4.98029136887077405513530143954, −4.07559221372765540296477349058, −3.23565811624305831734564096164, −2.21781143096155411582634059699, −1.01339434715343294662814590393, 0,
1.01339434715343294662814590393, 2.21781143096155411582634059699, 3.23565811624305831734564096164, 4.07559221372765540296477349058, 4.98029136887077405513530143954, 5.58344756264802333727634053775, 6.75904741051958840400035550399, 7.01812692974366469345650590427, 7.75433809423225303178239940813