| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 3·9-s − 10-s + 4·11-s + 4·14-s + 16-s + 3·17-s − 3·18-s − 20-s + 4·22-s − 4·23-s − 4·25-s + 4·28-s − 29-s + 4·31-s + 32-s + 3·34-s − 4·35-s − 3·36-s + 3·37-s − 40-s − 9·41-s − 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 9-s − 0.316·10-s + 1.20·11-s + 1.06·14-s + 1/4·16-s + 0.727·17-s − 0.707·18-s − 0.223·20-s + 0.852·22-s − 0.834·23-s − 4/5·25-s + 0.755·28-s − 0.185·29-s + 0.718·31-s + 0.176·32-s + 0.514·34-s − 0.676·35-s − 1/2·36-s + 0.493·37-s − 0.158·40-s − 1.40·41-s − 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.086772637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.086772637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−11.73393752866214395062941923882, −11.07718094423656610545951619270, −9.777318883006220542105164078322, −8.389455460221424173843846489864, −7.911995312311956948638277141106, −6.56150790811441553407226286731, −5.49349708810384813454586687760, −4.50196070453734734936420698072, −3.41530945529539786727500876542, −1.72818916811260562211346322788,
1.72818916811260562211346322788, 3.41530945529539786727500876542, 4.50196070453734734936420698072, 5.49349708810384813454586687760, 6.56150790811441553407226286731, 7.911995312311956948638277141106, 8.389455460221424173843846489864, 9.777318883006220542105164078322, 11.07718094423656610545951619270, 11.73393752866214395062941923882
