| L(s) = 1 | + 2.79·2-s + 5.79·4-s + 10.5·8-s − 3·9-s + 2.58·11-s + 17.9·16-s − 8.37·18-s + 7.20·22-s − 0.582·23-s − 10.1·29-s + 28.9·32-s − 17.3·36-s + 6.16·37-s − 10.5·43-s + 14.9·44-s − 1.62·46-s + 10·53-s − 28.3·58-s + 44.9·64-s − 11.7·67-s − 12.5·71-s − 31.7·72-s + 17.2·74-s − 17.7·79-s + 9·81-s − 29.5·86-s + 27.3·88-s + ⋯ |
| L(s) = 1 | + 1.97·2-s + 2.89·4-s + 3.74·8-s − 9-s + 0.778·11-s + 4.48·16-s − 1.97·18-s + 1.53·22-s − 0.121·23-s − 1.88·29-s + 5.11·32-s − 2.89·36-s + 1.01·37-s − 1.61·43-s + 2.25·44-s − 0.239·46-s + 1.37·53-s − 3.72·58-s + 5.61·64-s − 1.43·67-s − 1.49·71-s − 3.74·72-s + 2.00·74-s − 1.99·79-s + 81-s − 3.18·86-s + 2.91·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.697637275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.697637275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 11 | \( 1 - 2.58T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 0.582T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6.16T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 17.7T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.01959184322465980084712559146, −8.806359948205660945161948872373, −7.70283930173604069466681300658, −6.92726841305019761257651557534, −6.01411625363100731726211490944, −5.54831764284406831054848586852, −4.50964608075976483409271282677, −3.68350998539542172005763398597, −2.85543615097272829993642620410, −1.73895422661891684138330226755,
1.73895422661891684138330226755, 2.85543615097272829993642620410, 3.68350998539542172005763398597, 4.50964608075976483409271282677, 5.54831764284406831054848586852, 6.01411625363100731726211490944, 6.92726841305019761257651557534, 7.70283930173604069466681300658, 8.806359948205660945161948872373, 10.01959184322465980084712559146
