| L(s) = 1 | − 3i·3-s + (−10 − 5i)5-s + 10i·7-s − 9·9-s − 46·11-s + 34i·13-s + (−15 + 30i)15-s + 66i·17-s − 104·19-s + 30·21-s − 164i·23-s + (75 + 100i)25-s + 27i·27-s − 224·29-s − 72·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.894 − 0.447i)5-s + 0.539i·7-s − 0.333·9-s − 1.26·11-s + 0.725i·13-s + (−0.258 + 0.516i)15-s + 0.941i·17-s − 1.25·19-s + 0.311·21-s − 1.48i·23-s + (0.599 + 0.800i)25-s + 0.192i·27-s − 1.43·29-s − 0.417·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 5 | \( 1 + (10 + 5i)T \) |
| good | 7 | \( 1 - 10iT - 343T^{2} \) |
| 11 | \( 1 + 46T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 66iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 104T + 6.85e3T^{2} \) |
| 23 | \( 1 + 164iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 224T + 2.43e4T^{2} \) |
| 31 | \( 1 + 72T + 2.97e4T^{2} \) |
| 37 | \( 1 + 22iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 194T + 6.89e4T^{2} \) |
| 43 | \( 1 + 108iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 480iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 286iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 426T + 2.05e5T^{2} \) |
| 61 | \( 1 - 698T + 2.26e5T^{2} \) |
| 67 | \( 1 - 328iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 188T + 3.57e5T^{2} \) |
| 73 | \( 1 - 740iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 412iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3iT - 9.12e5T^{2} \) |
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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
−12.65682500324660206166449716274, −11.48940014617112979277166926438, −10.51552337640630528027926206114, −8.809327550622403946182273586564, −8.166901377964906849844234471551, −6.95631103962713017950960523055, −5.55531328262322641620521993255, −4.11131109584874450519352108659, −2.23215940033862635561626329548, 0,
2.94179454888193875562430667260, 4.21276455999393334017156306189, 5.55542996319191192670459700853, 7.29610982458847357039143586029, 8.032976968043496686889291818206, 9.506714390261479503630820251213, 10.70284126557561858924728749851, 11.17132808459488361828960735804, 12.57887120014640073013862625935
