| L(s) = 1 | − 1.23·3-s − 8.71·5-s + 33.7·7-s − 25.4·9-s + 57.0·11-s + 84.2·13-s + 10.7·15-s − 17·17-s + 159.·19-s − 41.7·21-s − 57.9·23-s − 49.1·25-s + 64.8·27-s − 124.·29-s − 98.3·31-s − 70.5·33-s − 294.·35-s − 355.·37-s − 104.·39-s − 67.6·41-s + 218.·43-s + 221.·45-s + 312.·47-s + 798.·49-s + 21.0·51-s + 148.·53-s − 497.·55-s + ⋯ |
| L(s) = 1 | − 0.237·3-s − 0.779·5-s + 1.82·7-s − 0.943·9-s + 1.56·11-s + 1.79·13-s + 0.185·15-s − 0.242·17-s + 1.92·19-s − 0.433·21-s − 0.525·23-s − 0.392·25-s + 0.462·27-s − 0.798·29-s − 0.569·31-s − 0.371·33-s − 1.42·35-s − 1.57·37-s − 0.427·39-s − 0.257·41-s + 0.775·43-s + 0.735·45-s + 0.969·47-s + 2.32·49-s + 0.0576·51-s + 0.385·53-s − 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.447866023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.447866023\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 + 1.23T + 27T^{2} \) |
| 5 | \( 1 + 8.71T + 125T^{2} \) |
| 7 | \( 1 - 33.7T + 343T^{2} \) |
| 11 | \( 1 - 57.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.2T + 2.19e3T^{2} \) |
| 19 | \( 1 - 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 57.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 98.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 355.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 67.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 218.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 312.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 148.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 7.67T + 2.05e5T^{2} \) |
| 61 | \( 1 - 456.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 47.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 786.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 670.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 34.1T + 4.93e5T^{2} \) |
| 83 | \( 1 + 322.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 417.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 996.T + 9.12e5T^{2} \) |
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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
−9.150961945145009847676007012045, −8.654281290249263159531625407524, −7.925029407494526461814916496134, −7.13211270864112558050355428385, −5.92227860104754384461930727790, −5.28583083510867047514422669564, −4.07286522185619632710804768884, −3.53170346250484342129856555588, −1.74766554270114953734135144693, −0.911624807191365775059581764675,
0.911624807191365775059581764675, 1.74766554270114953734135144693, 3.53170346250484342129856555588, 4.07286522185619632710804768884, 5.28583083510867047514422669564, 5.92227860104754384461930727790, 7.13211270864112558050355428385, 7.925029407494526461814916496134, 8.654281290249263159531625407524, 9.150961945145009847676007012045
