L(s) = 1 | + 0.567·2-s + 3-s − 1.67·4-s + (−2.10 − 0.760i)5-s + 0.567·6-s + (2.63 − 0.228i)7-s − 2.08·8-s + 9-s + (−1.19 − 0.431i)10-s + (−0.727 + 3.23i)11-s − 1.67·12-s + 4.44i·13-s + (1.49 − 0.129i)14-s + (−2.10 − 0.760i)15-s + 2.17·16-s − 3.87i·17-s + ⋯ |
L(s) = 1 | + 0.401·2-s + 0.577·3-s − 0.839·4-s + (−0.940 − 0.340i)5-s + 0.231·6-s + (0.996 − 0.0863i)7-s − 0.737·8-s + 0.333·9-s + (−0.377 − 0.136i)10-s + (−0.219 + 0.975i)11-s − 0.484·12-s + 1.23i·13-s + (0.399 − 0.0346i)14-s + (−0.542 − 0.196i)15-s + 0.543·16-s − 0.939i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.557471708\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.557471708\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + (2.10 + 0.760i)T \) |
| 7 | \( 1 + (-2.63 + 0.228i)T \) |
| 11 | \( 1 + (0.727 - 3.23i)T \) |
good | 2 | \( 1 - 0.567T + 2T^{2} \) |
| 13 | \( 1 - 4.44iT - 13T^{2} \) |
| 17 | \( 1 + 3.87iT - 17T^{2} \) |
| 19 | \( 1 + 0.293T + 19T^{2} \) |
| 23 | \( 1 - 1.67iT - 23T^{2} \) |
| 29 | \( 1 - 0.242iT - 29T^{2} \) |
| 31 | \( 1 - 6.28iT - 31T^{2} \) |
| 37 | \( 1 - 11.3iT - 37T^{2} \) |
| 41 | \( 1 - 5.68T + 41T^{2} \) |
| 43 | \( 1 - 7.05T + 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 5.48iT - 59T^{2} \) |
| 61 | \( 1 + 5.57T + 61T^{2} \) |
| 67 | \( 1 - 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 4.76iT - 73T^{2} \) |
| 79 | \( 1 + 7.37iT - 79T^{2} \) |
| 83 | \( 1 - 12.9iT - 83T^{2} \) |
| 89 | \( 1 + 13.0iT - 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−9.637054134678971936097067484677, −9.075564541584881236343521310364, −8.311488411081444980881328476330, −7.61336382757920471925943466301, −6.82525946715101652476263667448, −5.23713207682194901816389371694, −4.55351237720500788844178042317, −4.13210992438303101640330263970, −2.87368692548928599579763993153, −1.36532575948311427458838046406,
0.63592562264623836114712635073, 2.57451593155052820805090146885, 3.63738803494885136057694215811, 4.20084025271295729784214335890, 5.29019393817579773727764959356, 6.06163715273360675065643574322, 7.55994788080860673815375571445, 8.127860821030371714066539824456, 8.532668769672831195603353462109, 9.507148092675446812697653202992