| L(s) = 1 | − i·2-s − 1.92i·3-s − 4-s + (2.03 + 0.931i)5-s − 1.92·6-s − 1.36i·7-s + i·8-s − 0.713·9-s + (0.931 − 2.03i)10-s + 5.21·11-s + 1.92i·12-s − 1.36·14-s + (1.79 − 3.91i)15-s + 16-s − 1.48i·17-s + 0.713i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.11i·3-s − 0.5·4-s + (0.909 + 0.416i)5-s − 0.786·6-s − 0.515i·7-s + 0.353i·8-s − 0.237·9-s + (0.294 − 0.642i)10-s + 1.57·11-s + 0.556i·12-s − 0.364·14-s + (0.463 − 1.01i)15-s + 0.250·16-s − 0.359i·17-s + 0.168i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.238281835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.238281835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (-2.03 - 0.931i)T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.92iT - 3T^{2} \) |
| 7 | \( 1 + 1.36iT - 7T^{2} \) |
| 11 | \( 1 - 5.21T + 11T^{2} \) |
| 17 | \( 1 + 1.48iT - 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 - 6.67iT - 23T^{2} \) |
| 29 | \( 1 - 5.96T + 29T^{2} \) |
| 31 | \( 1 - 7.98T + 31T^{2} \) |
| 37 | \( 1 + 4.68iT - 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 - 10.9iT - 43T^{2} \) |
| 47 | \( 1 + 0.284iT - 47T^{2} \) |
| 53 | \( 1 + 13.2iT - 53T^{2} \) |
| 59 | \( 1 + 4.66T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 + 8.97iT - 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 6.17iT - 73T^{2} \) |
| 79 | \( 1 - 9.84T + 79T^{2} \) |
| 83 | \( 1 - 6.94iT - 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 10.1iT - 97T^{2} \) |
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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
−9.328107779850371638902216315385, −8.327105257669290354404492693106, −7.39625297291679589037244521908, −6.60107543656139077530934113736, −6.18351866352154790726682369181, −4.89894990194036561906581921376, −3.85052505317425440952082889851, −2.79973790566790171695829587005, −1.69588430174620927476555953968, −1.07742269273263193630485407048,
1.30624407024964647346227829307, 2.80631531155062231617516035919, 4.19152235237824404106204982652, 4.55359099218138137461319027093, 5.56068987563600474839853893354, 6.33082804134097574808867909410, 6.89690227518965467104582052900, 8.544110002751916355456538889460, 8.690648742259787069695648011301, 9.518622368442604173646158848509
