| L(s) = 1 | − 0.568·3-s − 5-s + 4.73·7-s − 2.67·9-s − 0.360·11-s + 5.26·13-s + 0.568·15-s + 0.370·17-s − 4.60·19-s − 2.69·21-s − 23-s + 25-s + 3.22·27-s + 0.939·29-s + 9.66·31-s + 0.204·33-s − 4.73·35-s + 3.26·37-s − 2.99·39-s + 5.29·41-s + 2.67·45-s − 1.25·47-s + 15.4·49-s − 0.210·51-s + 10.9·53-s + 0.360·55-s + 2.61·57-s + ⋯ |
| L(s) = 1 | − 0.328·3-s − 0.447·5-s + 1.79·7-s − 0.892·9-s − 0.108·11-s + 1.45·13-s + 0.146·15-s + 0.0899·17-s − 1.05·19-s − 0.587·21-s − 0.208·23-s + 0.200·25-s + 0.620·27-s + 0.174·29-s + 1.73·31-s + 0.0356·33-s − 0.800·35-s + 0.537·37-s − 0.478·39-s + 0.827·41-s + 0.399·45-s − 0.183·47-s + 2.20·49-s − 0.0295·51-s + 1.50·53-s + 0.0486·55-s + 0.346·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.547256432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.547256432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 0.568T + 3T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 0.360T + 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 - 0.370T + 17T^{2} \) |
| 19 | \( 1 + 4.60T + 19T^{2} \) |
| 29 | \( 1 - 0.939T + 29T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 1.25T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 9.66T + 59T^{2} \) |
| 61 | \( 1 - 9.71T + 61T^{2} \) |
| 67 | \( 1 + 7.07T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 0.745T + 73T^{2} \) |
| 79 | \( 1 - 0.415T + 79T^{2} \) |
| 83 | \( 1 + 9.26T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
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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
−10.45199657384539570489724629381, −8.921526844387388324171576069321, −8.300725025936075732036831132258, −7.889655856403614583138782715275, −6.52939515855483613126762930244, −5.71377712578460314387666017935, −4.75616832661905704282225457331, −3.95100545106586931437065464553, −2.48189551919500866987083061786, −1.07702969525552984457780324255,
1.07702969525552984457780324255, 2.48189551919500866987083061786, 3.95100545106586931437065464553, 4.75616832661905704282225457331, 5.71377712578460314387666017935, 6.52939515855483613126762930244, 7.889655856403614583138782715275, 8.300725025936075732036831132258, 8.921526844387388324171576069321, 10.45199657384539570489724629381
