| L(s) = 1 | + 0.493i·3-s + (1.09 + 1.95i)5-s − 4.54i·7-s + 2.75·9-s − 4.61·11-s − 5.54i·13-s + (−0.963 + 0.538i)15-s − 7.16i·17-s − 1.35·19-s + 2.24·21-s − i·23-s + (−2.62 + 4.25i)25-s + 2.84i·27-s + 3.66·29-s − 4.46·31-s + ⋯ |
| L(s) = 1 | + 0.284i·3-s + (0.487 + 0.872i)5-s − 1.71i·7-s + 0.918·9-s − 1.39·11-s − 1.53i·13-s + (−0.248 + 0.138i)15-s − 1.73i·17-s − 0.311·19-s + 0.489·21-s − 0.208i·23-s + (−0.524 + 0.851i)25-s + 0.546i·27-s + 0.680·29-s − 0.802·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29541 - 0.760103i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29541 - 0.760103i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.09 - 1.95i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 0.493iT - 3T^{2} \) |
| 7 | \( 1 + 4.54iT - 7T^{2} \) |
| 11 | \( 1 + 4.61T + 11T^{2} \) |
| 13 | \( 1 + 5.54iT - 13T^{2} \) |
| 17 | \( 1 + 7.16iT - 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 29 | \( 1 - 3.66T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 3.32iT - 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 - 8.68iT - 43T^{2} \) |
| 47 | \( 1 + 9.59iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 - 2.08T + 59T^{2} \) |
| 61 | \( 1 + 0.686T + 61T^{2} \) |
| 67 | \( 1 + 8.84iT - 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 - 4.44iT - 73T^{2} \) |
| 79 | \( 1 - 8.65T + 79T^{2} \) |
| 83 | \( 1 + 4.43iT - 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 1.21iT - 97T^{2} \) |
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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
−10.09314233227010804287681342623, −9.516927802345416724536338661343, −7.84160779425613226169548746671, −7.47485310486039748682843172731, −6.73837940707105019560340949081, −5.46549487327842845217509541198, −4.62268327642009038040515053250, −3.47767761305605488956825737111, −2.56534371717042132370411480631, −0.70936609504300154655545125035,
1.73062528074813721344556672454, 2.34226709096507556304382769051, 4.10306584882352276051268379523, 5.07353419126701956768884082428, 5.85221230260125400215121578182, 6.62940368259969221662859671923, 7.948789154682311407663619458053, 8.575771052910472818813160301182, 9.301080898392869686264131442933, 10.04814631579991297263473556858
