| L(s) = 1 | − 1.76i·2-s − i·3-s − 1.13·4-s + (−1.09 + 1.94i)5-s − 1.76·6-s + i·7-s − 1.53i·8-s − 9-s + (3.44 + 1.93i)10-s − 11-s + 1.13i·12-s − 2.21i·13-s + 1.76·14-s + (1.94 + 1.09i)15-s − 4.98·16-s + 7.13i·17-s + ⋯ |
| L(s) = 1 | − 1.25i·2-s − 0.577i·3-s − 0.565·4-s + (−0.489 + 0.871i)5-s − 0.722·6-s + 0.377i·7-s − 0.543i·8-s − 0.333·9-s + (1.09 + 0.612i)10-s − 0.301·11-s + 0.326i·12-s − 0.615i·13-s + 0.472·14-s + (0.503 + 0.282i)15-s − 1.24·16-s + 1.73i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2921856782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2921856782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 + (1.09 - 1.94i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.76iT - 2T^{2} \) |
| 13 | \( 1 + 2.21iT - 13T^{2} \) |
| 17 | \( 1 - 7.13iT - 17T^{2} \) |
| 19 | \( 1 + 7.91T + 19T^{2} \) |
| 23 | \( 1 - 1.96iT - 23T^{2} \) |
| 29 | \( 1 + 5.96T + 29T^{2} \) |
| 31 | \( 1 + 1.40T + 31T^{2} \) |
| 37 | \( 1 - 1.37iT - 37T^{2} \) |
| 41 | \( 1 - 4.80T + 41T^{2} \) |
| 43 | \( 1 - 2.34iT - 43T^{2} \) |
| 47 | \( 1 - 9.10iT - 47T^{2} \) |
| 53 | \( 1 - 2.65iT - 53T^{2} \) |
| 59 | \( 1 + 6.97T + 59T^{2} \) |
| 61 | \( 1 + 1.37T + 61T^{2} \) |
| 67 | \( 1 + 12.8iT - 67T^{2} \) |
| 71 | \( 1 - 1.16T + 71T^{2} \) |
| 73 | \( 1 - 4.94iT - 73T^{2} \) |
| 79 | \( 1 + 0.590T + 79T^{2} \) |
| 83 | \( 1 - 11.2iT - 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 + 12.8iT - 97T^{2} \) |
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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
−10.34093022251524331723386338666, −9.285918022833138655479057856517, −8.266733303104854470277017846627, −7.60457327709534530771566271713, −6.50257661477750017884464677997, −5.93262773018149969549633357568, −4.30196524196503862205657833863, −3.47351140620896419229291943484, −2.54265037146672774663978073449, −1.69986064082182004813237235273,
0.11722955269648599342360078435, 2.27730501150165928181133481970, 3.90735104178632967186605756484, 4.69897177864049910902478570547, 5.31932404359394357155734511377, 6.35962939009201268633174163792, 7.24505060163874364280417142215, 7.889701013977492531086240558347, 8.870967876182031712883047848607, 9.162209638177711889175157239247
