L(s) = 1 | + (−0.5 + 0.866i)3-s + (2 + 3.46i)5-s + (−0.499 − 0.866i)9-s + (−1 + 1.73i)11-s + 6·13-s − 3.99·15-s + (−2 + 3.46i)17-s + (−2 − 3.46i)19-s + (−1 − 1.73i)23-s + (−5.49 + 9.52i)25-s + 0.999·27-s − 2·29-s + (−0.999 − 1.73i)33-s + (−1 − 1.73i)37-s + (−3 + 5.19i)39-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.894 + 1.54i)5-s + (−0.166 − 0.288i)9-s + (−0.301 + 0.522i)11-s + 1.66·13-s − 1.03·15-s + (−0.485 + 0.840i)17-s + (−0.458 − 0.794i)19-s + (−0.208 − 0.361i)23-s + (−1.09 + 1.90i)25-s + 0.192·27-s − 0.371·29-s + (−0.174 − 0.301i)33-s + (−0.164 − 0.284i)37-s + (−0.480 + 0.832i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.899721 + 1.18266i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.899721 + 1.18266i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6 + 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.83553250140472189725151166527, −10.31919960158553618696624186020, −9.393406871908638830482608094938, −8.438036275958671216387069985601, −7.10718021291927976537269469928, −6.33009998602463648975077156395, −5.73306790284993074620557280429, −4.26749392543257546348208727425, −3.19062712664587901307123999110, −2.00116482784921729319586394075,
0.907627878263744397577862395914, 2.01887921111433387468975441929, 3.82794037451527102611794146884, 5.12712483830821929972538849272, 5.76722286330627082202851542354, 6.59240410576927878018585832543, 8.070712938542017047973833027263, 8.632588966772118602689620141827, 9.399908899577025617190880326426, 10.45008187374833463644346171798