| L(s) = 1 | − 2.13·2-s + 3-s + 2.55·4-s − 2.13·6-s + 2.16·7-s − 1.17·8-s + 9-s − 2.50·11-s + 2.55·12-s − 4.33·13-s − 4.62·14-s − 2.59·16-s − 6.77·17-s − 2.13·18-s + 6.83·19-s + 2.16·21-s + 5.35·22-s − 1.67·23-s − 1.17·24-s + 9.24·26-s + 27-s + 5.53·28-s + 4.44·29-s + 6.56·31-s + 7.88·32-s − 2.50·33-s + 14.4·34-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 0.577·3-s + 1.27·4-s − 0.870·6-s + 0.819·7-s − 0.415·8-s + 0.333·9-s − 0.756·11-s + 0.736·12-s − 1.20·13-s − 1.23·14-s − 0.648·16-s − 1.64·17-s − 0.502·18-s + 1.56·19-s + 0.473·21-s + 1.14·22-s − 0.349·23-s − 0.239·24-s + 1.81·26-s + 0.192·27-s + 1.04·28-s + 0.826·29-s + 1.17·31-s + 1.39·32-s − 0.436·33-s + 2.47·34-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
| \(L(1)\) |
\(\approx\) |
\(0.9607817628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9607817628\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.13T + 2T^{2} \) |
| 7 | \( 1 - 2.16T + 7T^{2} \) |
| 11 | \( 1 + 2.50T + 11T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 17 | \( 1 + 6.77T + 17T^{2} \) |
| 19 | \( 1 - 6.83T + 19T^{2} \) |
| 23 | \( 1 + 1.67T + 23T^{2} \) |
| 29 | \( 1 - 4.44T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 - 7.97T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 4.25T + 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 + 8.21T + 53T^{2} \) |
| 59 | \( 1 - 3.67T + 59T^{2} \) |
| 61 | \( 1 - 4.93T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 + 1.11T + 73T^{2} \) |
| 79 | \( 1 - 7.78T + 79T^{2} \) |
| 83 | \( 1 - 9.87T + 83T^{2} \) |
| 89 | \( 1 + 1.24T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
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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
−9.310913122342366031218370269997, −8.361034951201508646716352513192, −7.87922690760398073296826028834, −7.34597817625979480403994668137, −6.44831938806451573716260493081, −5.02806357356023258050533318766, −4.41580639633890287123486468101, −2.72658678122537410509583383682, −2.14156886220949796045974786182, −0.806568428686751634628426558246,
0.806568428686751634628426558246, 2.14156886220949796045974786182, 2.72658678122537410509583383682, 4.41580639633890287123486468101, 5.02806357356023258050533318766, 6.44831938806451573716260493081, 7.34597817625979480403994668137, 7.87922690760398073296826028834, 8.361034951201508646716352513192, 9.310913122342366031218370269997
