| L(s) = 1 | + i·2-s − 4-s + (1.67 + 1.48i)5-s − 1.19i·7-s − i·8-s + (−1.48 + 1.67i)10-s − 2·11-s + i·13-s + 1.19·14-s + 16-s + 4.54i·17-s − 4.15·19-s + (−1.67 − 1.48i)20-s − 2i·22-s + 7.11i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.749 + 0.662i)5-s − 0.451i·7-s − 0.353i·8-s + (−0.468 + 0.529i)10-s − 0.603·11-s + 0.277i·13-s + 0.319·14-s + 0.250·16-s + 1.10i·17-s − 0.953·19-s + (−0.374 − 0.331i)20-s − 0.426i·22-s + 1.48i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.385165832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.385165832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.67 - 1.48i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 + 1.19iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 4.54iT - 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 - 7.11iT - 23T^{2} \) |
| 29 | \( 1 - 10.7T + 29T^{2} \) |
| 31 | \( 1 + 5.35T + 31T^{2} \) |
| 37 | \( 1 - 3.92iT - 37T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 - 10.8iT - 43T^{2} \) |
| 47 | \( 1 + 1.61iT - 47T^{2} \) |
| 53 | \( 1 + 4.18iT - 53T^{2} \) |
| 59 | \( 1 + 2.31T + 59T^{2} \) |
| 61 | \( 1 + 7.08T + 61T^{2} \) |
| 67 | \( 1 - 4.70iT - 67T^{2} \) |
| 71 | \( 1 - 9.27T + 71T^{2} \) |
| 73 | \( 1 + 3.58iT - 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 1.73iT - 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 9.19iT - 97T^{2} \) |
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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
−10.15331073648043388827079556744, −9.253465142121131553566013293366, −8.329761195402655914272559429820, −7.55835588383092890203189883191, −6.65740643911797505735500673879, −6.09020036123218662411855144244, −5.15516370501241418437467989610, −4.10164632192913681471296729668, −2.98523700247761674794036627095, −1.61846595725382923159618177305,
0.59569675783565761835236191891, 2.14131810951456483567287302779, 2.81781449182612074086500396360, 4.33768758611873188842958745859, 5.06835332678259619329547873917, 5.87161091555660559148178774033, 6.90431656660370579667247754443, 8.199298993207656235020206543649, 8.782518858080373928462867965156, 9.460903939366425349065129410599
