L(s) = 1 | + i·2-s + (0.794 − 0.794i)3-s − 4-s + (0.217 + 2.22i)5-s + (0.794 + 0.794i)6-s + (2.93 − 2.93i)7-s − i·8-s + 1.73i·9-s + (−2.22 + 0.217i)10-s + 3.55i·11-s + (−0.794 + 0.794i)12-s − 4.41i·13-s + (2.93 + 2.93i)14-s + (1.94 + 1.59i)15-s + 16-s + 5.37·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (0.458 − 0.458i)3-s − 0.5·4-s + (0.0971 + 0.995i)5-s + (0.324 + 0.324i)6-s + (1.11 − 1.11i)7-s − 0.353i·8-s + 0.578i·9-s + (−0.703 + 0.0686i)10-s + 1.07i·11-s + (−0.229 + 0.229i)12-s − 1.22i·13-s + (0.785 + 0.785i)14-s + (0.501 + 0.412i)15-s + 0.250·16-s + 1.30·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48781 + 0.755924i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48781 + 0.755924i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-0.217 - 2.22i)T \) |
| 37 | \( 1 + (4.67 - 3.88i)T \) |
good | 3 | \( 1 + (-0.794 + 0.794i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.93 + 2.93i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.55iT - 11T^{2} \) |
| 13 | \( 1 + 4.41iT - 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 + (1.98 - 1.98i)T - 19iT^{2} \) |
| 23 | \( 1 - 5.90iT - 23T^{2} \) |
| 29 | \( 1 + (2.18 + 2.18i)T + 29iT^{2} \) |
| 31 | \( 1 + (-5.42 + 5.42i)T - 31iT^{2} \) |
| 41 | \( 1 + 6.34iT - 41T^{2} \) |
| 43 | \( 1 - 1.78iT - 43T^{2} \) |
| 47 | \( 1 + (-1.82 + 1.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.57 + 9.57i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.44 - 4.44i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.44 + 1.44i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.88 + 7.88i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + (-4.02 + 4.02i)T - 73iT^{2} \) |
| 79 | \( 1 + (2.03 - 2.03i)T - 79iT^{2} \) |
| 83 | \( 1 + (4.51 + 4.51i)T + 83iT^{2} \) |
| 89 | \( 1 + (2.13 + 2.13i)T + 89iT^{2} \) |
| 97 | \( 1 + 5.91T + 97T^{2} \) |
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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
−11.42592022982098322459507538749, −10.36677448748956896114210148656, −9.948613880209405371882262143282, −8.188074261009413899277466373407, −7.59883870419047302249199094075, −7.27902427345447695944323250950, −5.82325951665674469315350333018, −4.70978568958323623379413601613, −3.38575801607964049640987970707, −1.73592808571019692886988627564,
1.38698989164421709151728534743, 2.85345777455880806090813447733, 4.22477653906105528751997094990, 5.07504211508933736118955587421, 6.16197861415260441182273714228, 8.088984869605239518600729389073, 8.882191301441705074795748768186, 9.065431770419312220735847412435, 10.32848357556691848702967343624, 11.47092284352588172835895958958