| L(s) = 1 | + 0.540i·5-s + 1.23·7-s + 2.47·11-s − 4.57i·13-s + 3.36i·17-s − 1.74i·19-s − 8.61i·23-s + 4.70·25-s + 7.94i·29-s − 6.32i·31-s + 0.667i·35-s + (−2.23 − 5.65i)37-s − 2·41-s − 2.82i·43-s + 4·47-s + ⋯ |
| L(s) = 1 | + 0.241i·5-s + 0.467·7-s + 0.745·11-s − 1.26i·13-s + 0.817i·17-s − 0.401i·19-s − 1.79i·23-s + 0.941·25-s + 1.47i·29-s − 1.13i·31-s + 0.112i·35-s + (−0.367 − 0.929i)37-s − 0.312·41-s − 0.431i·43-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.923168196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.923168196\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (2.23 + 5.65i)T \) |
| good | 5 | \( 1 - 0.540iT - 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 + 4.57iT - 13T^{2} \) |
| 17 | \( 1 - 3.36iT - 17T^{2} \) |
| 19 | \( 1 + 1.74iT - 19T^{2} \) |
| 23 | \( 1 + 8.61iT - 23T^{2} \) |
| 29 | \( 1 - 7.94iT - 29T^{2} \) |
| 31 | \( 1 + 6.32iT - 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 2.82iT - 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 2.28iT - 59T^{2} \) |
| 61 | \( 1 - 5.65iT - 61T^{2} \) |
| 67 | \( 1 - 7.70T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 + 9.56iT - 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 + 8.61iT - 89T^{2} \) |
| 97 | \( 1 - 13.7iT - 97T^{2} \) |
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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
−8.141531567167490593853840646099, −7.31785453109684594322747866656, −6.61654960079336109074139066420, −5.95556764502630114290514498569, −5.09973056035468004875529107441, −4.40619621260101854226819849963, −3.49405439811996168443988026298, −2.71461668321230136717715315945, −1.67079387137310440153767295792, −0.54150987573453156244915303246,
1.18682585185619962889424152924, 1.86588739498803326529987840661, 3.07458377140746897142825149665, 3.91789653886722751269097007251, 4.71106040146817367187824793357, 5.27327086332295005008639872001, 6.32152797520581935951835861019, 6.82891293123246841948919931292, 7.63807904712209696937384015517, 8.299677748738159581663857834509
