L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s + 2·11-s + 12-s − 6·13-s − 2·14-s + 16-s − 2·17-s + 18-s − 2·21-s + 2·22-s + 4·23-s + 24-s − 6·26-s + 27-s − 2·28-s − 8·31-s + 32-s + 2·33-s − 2·34-s + 36-s − 2·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s − 1.66·13-s − 0.534·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.436·21-s + 0.426·22-s + 0.834·23-s + 0.204·24-s − 1.17·26-s + 0.192·27-s − 0.377·28-s − 1.43·31-s + 0.176·32-s + 0.348·33-s − 0.342·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.769259248\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.769259248\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.92564501122466550661052269686, −12.39023149942860104609602897620, −11.14346521042731785816533268110, −9.868458945526617237244084557500, −9.040702798039804053791802466769, −7.47947512213883760565427163251, −6.64959962043719359186062300669, −5.13544146251726925083565635926, −3.79143046769365773246334925917, −2.45815773509881651106722995929,
2.45815773509881651106722995929, 3.79143046769365773246334925917, 5.13544146251726925083565635926, 6.64959962043719359186062300669, 7.47947512213883760565427163251, 9.040702798039804053791802466769, 9.868458945526617237244084557500, 11.14346521042731785816533268110, 12.39023149942860104609602897620, 12.92564501122466550661052269686