| L(s) = 1 | + (−1.70 + 0.292i)3-s + (2.82 − i)9-s − 2.24·11-s + 6i·17-s − 7.41i·19-s + 5·25-s + (−4.53 + 2.53i)27-s + (3.82 − 0.656i)33-s − 11.3i·41-s + 13.0i·43-s + 7·49-s + (−1.75 − 10.2i)51-s + (2.17 + 12.6i)57-s + 5.75·59-s − 3.89i·67-s + ⋯ |
| L(s) = 1 | + (−0.985 + 0.169i)3-s + (0.942 − 0.333i)9-s − 0.676·11-s + 1.45i·17-s − 1.70i·19-s + 25-s + (−0.872 + 0.487i)27-s + (0.666 − 0.114i)33-s − 1.76i·41-s + 1.99i·43-s + 49-s + (−0.246 − 1.43i)51-s + (0.287 + 1.67i)57-s + 0.749·59-s − 0.476i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.085741296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.085741296\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.70 - 0.292i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 7.41iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 11.3iT - 41T^{2} \) |
| 43 | \( 1 - 13.0iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 3.89iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16.9T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 - 18iT - 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−9.538407909438308146840667083388, −8.767847927821483106087731408408, −7.78623847053516114319153598729, −6.89499575932657193984639486690, −6.23297127685913254333642648180, −5.28692723780819475071541890946, −4.64959810647082639321990297670, −3.61499551832928788151115532132, −2.28627087359349405980466171931, −0.78536452291307508072185846962,
0.76240109721483658536465849226, 2.15966906921685753708941129092, 3.45985261840464444112397334356, 4.65438881540045841310452461826, 5.32343093131718659166729319095, 6.09333415768614978404245031059, 7.03166201314646117348058450039, 7.64526258213176741135796132922, 8.572921415957242870778719571077, 9.670648614113132841703767777274
