| L(s) = 1 | − 3-s − 2.37·5-s + 0.792·7-s + 9-s + 4.10·13-s + 2.37·15-s + 5.98·17-s + 4.25·19-s − 0.792·21-s − 2·23-s + 0.627·25-s − 27-s + 2.52·29-s − 7.37·31-s − 1.87·35-s + 5·37-s − 4.10·39-s + 5.69·41-s − 6.63·43-s − 2.37·45-s + 1.25·47-s − 6.37·49-s − 5.98·51-s + 13.1·53-s − 4.25·57-s + 6·59-s + 2.67·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.06·5-s + 0.299·7-s + 0.333·9-s + 1.13·13-s + 0.612·15-s + 1.45·17-s + 0.976·19-s − 0.172·21-s − 0.417·23-s + 0.125·25-s − 0.192·27-s + 0.468·29-s − 1.32·31-s − 0.317·35-s + 0.821·37-s − 0.657·39-s + 0.889·41-s − 1.01·43-s − 0.353·45-s + 0.183·47-s − 0.910·49-s − 0.838·51-s + 1.80·53-s − 0.563·57-s + 0.781·59-s + 0.342·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.431698393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.431698393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 0.792T + 7T^{2} \) |
| 13 | \( 1 - 4.10T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 2.52T + 29T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 2.67T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 + 0.744T + 71T^{2} \) |
| 73 | \( 1 + 7.42T + 73T^{2} \) |
| 79 | \( 1 - 5.84T + 79T^{2} \) |
| 83 | \( 1 - 8.51T + 83T^{2} \) |
| 89 | \( 1 + 6.37T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.924486176928166750573946410555, −7.54940525758687027722254419532, −6.75645750586436336855206539769, −5.77783318063172725971944167799, −5.41026017733266539024011508560, −4.34502901857529311038697218176, −3.75421478179796100712893305325, −3.04748811118965175659898430685, −1.57053078870850617630041393314, −0.69990658331821664158612709497,
0.69990658331821664158612709497, 1.57053078870850617630041393314, 3.04748811118965175659898430685, 3.75421478179796100712893305325, 4.34502901857529311038697218176, 5.41026017733266539024011508560, 5.77783318063172725971944167799, 6.75645750586436336855206539769, 7.54940525758687027722254419532, 7.924486176928166750573946410555
