L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 5·11-s − 12-s + 13-s + 14-s + 16-s + 17-s − 18-s + 21-s + 5·22-s − 6·23-s + 24-s − 5·25-s − 26-s − 27-s − 28-s − 10·29-s − 8·31-s − 32-s + 5·33-s − 34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.218·21-s + 1.06·22-s − 1.25·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s − 1.85·29-s − 1.43·31-s − 0.176·32-s + 0.870·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.91606193656124739003864976887, −10.09232308996604778849825434589, −9.363551266314125453468629303075, −8.025407693218525427980316031873, −7.43208854303745278135460909362, −6.10761877659726684085908706448, −5.37002895885674566259205833480, −3.72832421421558189975884042631, −2.12052920002859578562836029663, 0,
2.12052920002859578562836029663, 3.72832421421558189975884042631, 5.37002895885674566259205833480, 6.10761877659726684085908706448, 7.43208854303745278135460909362, 8.025407693218525427980316031873, 9.363551266314125453468629303075, 10.09232308996604778849825434589, 10.91606193656124739003864976887