| L(s) = 1 | + 2.35·2-s − 3-s + 3.52·4-s − 2.35·6-s + 3.48·7-s + 3.58·8-s + 9-s + 2.93·11-s − 3.52·12-s − 1.87·13-s + 8.18·14-s + 1.38·16-s + 6.78·17-s + 2.35·18-s − 2.94·19-s − 3.48·21-s + 6.89·22-s − 5.49·23-s − 3.58·24-s − 4.40·26-s − 27-s + 12.2·28-s + 2.55·29-s + 0.418·31-s − 3.92·32-s − 2.93·33-s + 15.9·34-s + ⋯ |
| L(s) = 1 | + 1.66·2-s − 0.577·3-s + 1.76·4-s − 0.959·6-s + 1.31·7-s + 1.26·8-s + 0.333·9-s + 0.883·11-s − 1.01·12-s − 0.520·13-s + 2.18·14-s + 0.345·16-s + 1.64·17-s + 0.554·18-s − 0.676·19-s − 0.759·21-s + 1.46·22-s − 1.14·23-s − 0.732·24-s − 0.864·26-s − 0.192·27-s + 2.32·28-s + 0.474·29-s + 0.0750·31-s − 0.694·32-s − 0.510·33-s + 2.73·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.612874836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.612874836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 7 | \( 1 - 3.48T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 + 1.87T + 13T^{2} \) |
| 17 | \( 1 - 6.78T + 17T^{2} \) |
| 19 | \( 1 + 2.94T + 19T^{2} \) |
| 23 | \( 1 + 5.49T + 23T^{2} \) |
| 29 | \( 1 - 2.55T + 29T^{2} \) |
| 31 | \( 1 - 0.418T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 - 1.67T + 41T^{2} \) |
| 43 | \( 1 - 10.9T + 43T^{2} \) |
| 47 | \( 1 + 7.49T + 47T^{2} \) |
| 53 | \( 1 - 3.70T + 53T^{2} \) |
| 59 | \( 1 + 7.10T + 59T^{2} \) |
| 61 | \( 1 + 6.43T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 0.728T + 71T^{2} \) |
| 73 | \( 1 + 3.59T + 73T^{2} \) |
| 79 | \( 1 - 3.07T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 0.287T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−9.342823986244490264060244218523, −8.085643136817928627258751235405, −7.49274928846248234224540947582, −6.44194527210376924931326287698, −5.84580528007292110922742536911, −5.07187141821216548034528920949, −4.41194512300348747292155035893, −3.72760112637302660053887752762, −2.44561084554159418761349840366, −1.35743388700795010778657274901,
1.35743388700795010778657274901, 2.44561084554159418761349840366, 3.72760112637302660053887752762, 4.41194512300348747292155035893, 5.07187141821216548034528920949, 5.84580528007292110922742536911, 6.44194527210376924931326287698, 7.49274928846248234224540947582, 8.085643136817928627258751235405, 9.342823986244490264060244218523
