| L(s) = 1 | + 0.226·2-s − 1.51·3-s − 1.94·4-s + 2.45·5-s − 0.344·6-s + 0.487·7-s − 0.895·8-s − 0.698·9-s + 0.556·10-s + 2.99·11-s + 2.95·12-s + 1.49·13-s + 0.110·14-s − 3.72·15-s + 3.69·16-s + 3.52·17-s − 0.158·18-s + 6.32·19-s − 4.78·20-s − 0.739·21-s + 0.679·22-s + 1.27·23-s + 1.35·24-s + 1.02·25-s + 0.338·26-s + 5.61·27-s − 0.949·28-s + ⋯ |
| L(s) = 1 | + 0.160·2-s − 0.875·3-s − 0.974·4-s + 1.09·5-s − 0.140·6-s + 0.184·7-s − 0.316·8-s − 0.232·9-s + 0.176·10-s + 0.902·11-s + 0.853·12-s + 0.413·13-s + 0.0295·14-s − 0.961·15-s + 0.923·16-s + 0.855·17-s − 0.0373·18-s + 1.45·19-s − 1.06·20-s − 0.161·21-s + 0.144·22-s + 0.265·23-s + 0.277·24-s + 0.205·25-s + 0.0663·26-s + 1.07·27-s − 0.179·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109979873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109979873\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 349 | \( 1 - T \) |
| good | 2 | \( 1 - 0.226T + 2T^{2} \) |
| 3 | \( 1 + 1.51T + 3T^{2} \) |
| 5 | \( 1 - 2.45T + 5T^{2} \) |
| 7 | \( 1 - 0.487T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 - 1.49T + 13T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 - 1.27T + 23T^{2} \) |
| 29 | \( 1 + 8.44T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 9.81T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 8.37T + 47T^{2} \) |
| 53 | \( 1 - 1.19T + 53T^{2} \) |
| 59 | \( 1 + 6.98T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 0.801T + 67T^{2} \) |
| 71 | \( 1 + 0.317T + 71T^{2} \) |
| 73 | \( 1 + 4.67T + 73T^{2} \) |
| 79 | \( 1 + 2.75T + 79T^{2} \) |
| 83 | \( 1 - 4.05T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−11.61494753513542587097700961866, −10.52946566021228486049001967345, −9.569553497296055416018121303243, −9.029474542950806853928200627906, −7.71861748562704470086890692079, −6.19637672103420649975330052841, −5.66574994387041652952269306536, −4.77245397379405253396573044706, −3.32678263161631353865499538314, −1.19191761626895597477803396242,
1.19191761626895597477803396242, 3.32678263161631353865499538314, 4.77245397379405253396573044706, 5.66574994387041652952269306536, 6.19637672103420649975330052841, 7.71861748562704470086890692079, 9.029474542950806853928200627906, 9.569553497296055416018121303243, 10.52946566021228486049001967345, 11.61494753513542587097700961866
