L(s) = 1 | + (−0.777 − 1.54i)3-s + 3.40i·5-s + (−1.79 + 2.40i)9-s − 3.94·13-s + (5.26 − 2.64i)15-s − 6.09i·17-s + 4.38i·19-s + 8.33·23-s − 6.58·25-s + (5.11 + 0.901i)27-s + 3.80i·29-s + 4.89i·31-s − 7.58·37-s + (3.06 + 6.10i)39-s + 0.710i·41-s + ⋯ |
L(s) = 1 | + (−0.448 − 0.893i)3-s + 1.52i·5-s + (−0.597 + 0.802i)9-s − 1.09·13-s + (1.36 − 0.683i)15-s − 1.47i·17-s + 1.00i·19-s + 1.73·23-s − 1.31·25-s + (0.984 + 0.173i)27-s + 0.707i·29-s + 0.879i·31-s − 1.24·37-s + (0.491 + 0.978i)39-s + 0.110i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1400207776\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1400207776\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.777 + 1.54i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.40iT - 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 3.94T + 13T^{2} \) |
| 17 | \( 1 + 6.09iT - 17T^{2} \) |
| 19 | \( 1 - 4.38iT - 19T^{2} \) |
| 23 | \( 1 - 8.33T + 23T^{2} \) |
| 29 | \( 1 - 3.80iT - 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 + 7.58T + 37T^{2} \) |
| 41 | \( 1 - 0.710iT - 41T^{2} \) |
| 43 | \( 1 + 9.66iT - 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 1.00iT - 53T^{2} \) |
| 59 | \( 1 + 4.66T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 6.59T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 6.09iT - 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−9.414787782050968279701357086906, −8.442003842453120843480590074108, −7.33626300063686018798684632370, −7.14206608412915332928609615029, −6.57596634571362579503972281288, −5.47612525214657235794339733349, −4.87233107934363217608942191435, −3.23239423167033405736791865589, −2.77399676771779220546015175981, −1.63441506154747344285367609335,
0.05127888894417966603979022133, 1.33574688500034873458905242142, 2.82666257941115630552499677494, 4.01807698845406068660930092428, 4.74850955216943236714950784959, 5.15831409102842920148404802238, 6.04000161236442355825431080525, 6.99492025473507840487022290824, 8.139495019604018336590420731851, 8.711958950549484455391953881400