| L(s) = 1 | + 2·2-s + 8·3-s + 4·4-s − 5·5-s + 16·6-s + 8·8-s + 37·9-s − 10·10-s + 12·11-s + 32·12-s + 58·13-s − 40·15-s + 16·16-s − 66·17-s + 74·18-s + 100·19-s − 20·20-s + 24·22-s + 132·23-s + 64·24-s + 25·25-s + 116·26-s + 80·27-s − 90·29-s − 80·30-s − 152·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 1/2·4-s − 0.447·5-s + 1.08·6-s + 0.353·8-s + 1.37·9-s − 0.316·10-s + 0.328·11-s + 0.769·12-s + 1.23·13-s − 0.688·15-s + 1/4·16-s − 0.941·17-s + 0.968·18-s + 1.20·19-s − 0.223·20-s + 0.232·22-s + 1.19·23-s + 0.544·24-s + 1/5·25-s + 0.874·26-s + 0.570·27-s − 0.576·29-s − 0.486·30-s − 0.880·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.247658093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.247658093\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 132 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 438 T + p^{3} T^{2} \) |
| 43 | \( 1 - 32 T + p^{3} T^{2} \) |
| 47 | \( 1 - 204 T + p^{3} T^{2} \) |
| 53 | \( 1 - 222 T + p^{3} T^{2} \) |
| 59 | \( 1 + 420 T + p^{3} T^{2} \) |
| 61 | \( 1 + 902 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1024 T + p^{3} T^{2} \) |
| 71 | \( 1 - 432 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 + 160 T + p^{3} T^{2} \) |
| 83 | \( 1 + 72 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1106 T + p^{3} T^{2} \) |
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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
−10.73658260281362148619536450940, −9.252623632791610704218499799601, −8.885961350295804892542642103262, −7.74974014986234195049779034566, −7.10609510711787411153545829952, −5.84234121340824604379404952862, −4.42121363232926495205334727802, −3.58659471138166665605015375794, −2.79652536489598075351163586538, −1.41968591770692404719310089490,
1.41968591770692404719310089490, 2.79652536489598075351163586538, 3.58659471138166665605015375794, 4.42121363232926495205334727802, 5.84234121340824604379404952862, 7.10609510711787411153545829952, 7.74974014986234195049779034566, 8.885961350295804892542642103262, 9.252623632791610704218499799601, 10.73658260281362148619536450940
