| L(s) = 1 | + 2.79·2-s − 3-s + 5.79·4-s + 3·5-s − 2.79·6-s − 7-s + 10.5·8-s + 9-s + 8.37·10-s − 5.79·12-s − 13-s − 2.79·14-s − 3·15-s + 17.9·16-s + 1.58·17-s + 2.79·18-s − 2.58·19-s + 17.3·20-s + 21-s + 3.58·23-s − 10.5·24-s + 4·25-s − 2.79·26-s − 27-s − 5.79·28-s − 10.1·29-s − 8.37·30-s + ⋯ |
| L(s) = 1 | + 1.97·2-s − 0.577·3-s + 2.89·4-s + 1.34·5-s − 1.13·6-s − 0.377·7-s + 3.74·8-s + 0.333·9-s + 2.64·10-s − 1.67·12-s − 0.277·13-s − 0.746·14-s − 0.774·15-s + 4.48·16-s + 0.383·17-s + 0.657·18-s − 0.592·19-s + 3.88·20-s + 0.218·21-s + 0.747·23-s − 2.16·24-s + 0.800·25-s − 0.547·26-s − 0.192·27-s − 1.09·28-s − 1.88·29-s − 1.52·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.934669386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.934669386\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.79T + 2T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 - 3.58T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 7.16T + 41T^{2} \) |
| 43 | \( 1 - 7.58T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 0.417T + 53T^{2} \) |
| 59 | \( 1 + 4.58T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 0.582T + 67T^{2} \) |
| 71 | \( 1 + 7.16T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 2.41T + 83T^{2} \) |
| 89 | \( 1 + 9.16T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−9.105707724808017798853333694717, −7.52872696872587286192296732135, −7.02294558392243815964767589542, −6.09684599860608992804544130679, −5.76094621349309937392180384782, −5.14604717568885430800938250250, −4.25240172795480316619529293514, −3.33146277758417999291721434520, −2.34606150932660084426562243744, −1.56929116736556763104516901440,
1.56929116736556763104516901440, 2.34606150932660084426562243744, 3.33146277758417999291721434520, 4.25240172795480316619529293514, 5.14604717568885430800938250250, 5.76094621349309937392180384782, 6.09684599860608992804544130679, 7.02294558392243815964767589542, 7.52872696872587286192296732135, 9.105707724808017798853333694717
