| L(s) = 1 | + 2.41·2-s + 1.41·3-s + 3.82·4-s + 3.41·6-s − 4.82·7-s + 4.41·8-s − 0.999·9-s + 3.41·11-s + 5.41·12-s + 13-s − 11.6·14-s + 2.99·16-s − 0.828·17-s − 2.41·18-s + 0.585·19-s − 6.82·21-s + 8.24·22-s − 1.41·23-s + 6.24·24-s + 2.41·26-s − 5.65·27-s − 18.4·28-s − 5.65·29-s + 1.75·31-s − 1.58·32-s + 4.82·33-s − 1.99·34-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 0.816·3-s + 1.91·4-s + 1.39·6-s − 1.82·7-s + 1.56·8-s − 0.333·9-s + 1.02·11-s + 1.56·12-s + 0.277·13-s − 3.11·14-s + 0.749·16-s − 0.200·17-s − 0.569·18-s + 0.134·19-s − 1.49·21-s + 1.75·22-s − 0.294·23-s + 1.27·24-s + 0.473·26-s − 1.08·27-s − 3.49·28-s − 1.05·29-s + 0.315·31-s − 0.280·32-s + 0.840·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.515462478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.515462478\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 3.41T + 11T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 - 0.585T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 2.48T + 53T^{2} \) |
| 59 | \( 1 - 1.75T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 7.65T + 97T^{2} \) |
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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
−11.98851720669363499114006441604, −11.03695148106214761386622810969, −9.637457415543295880970012135246, −8.965366051204050460810662430924, −7.40532028527185204058298052249, −6.36564015065318793199579584516, −5.81638356123165706209115181429, −4.11233001423733721128704766639, −3.43407149252504548147344905218, −2.52103154495935654149200801156,
2.52103154495935654149200801156, 3.43407149252504548147344905218, 4.11233001423733721128704766639, 5.81638356123165706209115181429, 6.36564015065318793199579584516, 7.40532028527185204058298052249, 8.965366051204050460810662430924, 9.637457415543295880970012135246, 11.03695148106214761386622810969, 11.98851720669363499114006441604
