L(s) = 1 | + 3·3-s + 16·5-s + 9·9-s + 18·11-s + 54·13-s + 48·15-s + 128·17-s + 52·19-s + 202·23-s + 131·25-s + 27·27-s + 302·29-s − 200·31-s + 54·33-s − 150·37-s + 162·39-s − 172·41-s − 164·43-s + 144·45-s − 460·47-s + 384·51-s − 190·53-s + 288·55-s + 156·57-s + 96·59-s − 622·61-s + 864·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.43·5-s + 1/3·9-s + 0.493·11-s + 1.15·13-s + 0.826·15-s + 1.82·17-s + 0.627·19-s + 1.83·23-s + 1.04·25-s + 0.192·27-s + 1.93·29-s − 1.15·31-s + 0.284·33-s − 0.666·37-s + 0.665·39-s − 0.655·41-s − 0.581·43-s + 0.477·45-s − 1.42·47-s + 1.05·51-s − 0.492·53-s + 0.706·55-s + 0.362·57-s + 0.211·59-s − 1.30·61-s + 1.64·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(5.206227783\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.206227783\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 18 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 128 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 - 202 T + p^{3} T^{2} \) |
| 29 | \( 1 - 302 T + p^{3} T^{2} \) |
| 31 | \( 1 + 200 T + p^{3} T^{2} \) |
| 37 | \( 1 + 150 T + p^{3} T^{2} \) |
| 41 | \( 1 + 172 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 460 T + p^{3} T^{2} \) |
| 53 | \( 1 + 190 T + p^{3} T^{2} \) |
| 59 | \( 1 - 96 T + p^{3} T^{2} \) |
| 61 | \( 1 + 622 T + p^{3} T^{2} \) |
| 67 | \( 1 + 744 T + p^{3} T^{2} \) |
| 71 | \( 1 - 54 T + p^{3} T^{2} \) |
| 73 | \( 1 + 742 T + p^{3} T^{2} \) |
| 79 | \( 1 - 92 T + p^{3} T^{2} \) |
| 83 | \( 1 + 228 T + p^{3} T^{2} \) |
| 89 | \( 1 - 116 T + p^{3} T^{2} \) |
| 97 | \( 1 - 554 T + p^{3} T^{2} \) |
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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
−8.835766319212669541318935516106, −7.972842055333310495224805100115, −7.00244209422800040426990217372, −6.30350548809670710781485941560, −5.52076913744339120229798253304, −4.81203723462899232920169892027, −3.38878870007776854472266138284, −2.99790958861822472936338456586, −1.55029132310867121298434170628, −1.18720429367725370322820570190,
1.18720429367725370322820570190, 1.55029132310867121298434170628, 2.99790958861822472936338456586, 3.38878870007776854472266138284, 4.81203723462899232920169892027, 5.52076913744339120229798253304, 6.30350548809670710781485941560, 7.00244209422800040426990217372, 7.972842055333310495224805100115, 8.835766319212669541318935516106