| L(s) = 1 | + i·3-s + (−0.353 − 2.20i)5-s − 3.09i·7-s − 9-s − 5.51·11-s + i·13-s + (2.20 − 0.353i)15-s + 6.21i·17-s + 3.09·21-s − 4.21i·23-s + (−4.74 + 1.56i)25-s − i·27-s + 1.70·29-s − 5.51i·33-s + (−6.83 + 1.09i)35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.158 − 0.987i)5-s − 1.16i·7-s − 0.333·9-s − 1.66·11-s + 0.277i·13-s + (0.570 − 0.0913i)15-s + 1.50i·17-s + 0.675·21-s − 0.879i·23-s + (−0.949 + 0.312i)25-s − 0.192i·27-s + 0.317·29-s − 0.959i·33-s + (−1.15 + 0.184i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6591299520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6591299520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.353 + 2.20i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 + 3.09iT - 7T^{2} \) |
| 11 | \( 1 + 5.51T + 11T^{2} \) |
| 17 | \( 1 - 6.21iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4.21iT - 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4.02iT - 37T^{2} \) |
| 41 | \( 1 - 0.198T + 41T^{2} \) |
| 43 | \( 1 - 5.70iT - 43T^{2} \) |
| 47 | \( 1 + 6.41iT - 47T^{2} \) |
| 53 | \( 1 - 4.02iT - 53T^{2} \) |
| 59 | \( 1 + 7.53T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 4.83iT - 67T^{2} \) |
| 71 | \( 1 + 7.98T + 71T^{2} \) |
| 73 | \( 1 - 9.31iT - 73T^{2} \) |
| 79 | \( 1 - 8.51T + 79T^{2} \) |
| 83 | \( 1 - 14.7iT - 83T^{2} \) |
| 89 | \( 1 - 9.40T + 89T^{2} \) |
| 97 | \( 1 + 14.3iT - 97T^{2} \) |
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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
−8.738589910775866032484805010074, −8.188593629209589241085243951934, −7.66148298874686436508471564608, −6.62028335349042304410717046351, −5.68773367131032109904761694149, −4.89769666884773485199101022056, −4.30340885097817407273187894169, −3.58341198009767310795105480982, −2.36248764995748573307530239664, −1.03921646776115361471717209972,
0.22447652391590633429317048679, 2.10065829592837118655949822045, 2.71039585267873763470712031107, 3.32873240645577672483565968809, 4.87490188059973926959457421079, 5.53197201788059060745264676589, 6.15654892356302709900595257116, 7.21322042925981245471906843017, 7.58172926218097402152333061883, 8.328598920458842964556354326138
