L(s) = 1 | − 2.74·2-s + 2.37·3-s + 5.53·4-s − 6.50·6-s + 0.409·7-s − 9.71·8-s + 2.62·9-s + 5.72·11-s + 13.1·12-s − 1.36·13-s − 1.12·14-s + 15.5·16-s − 3.94·17-s − 7.19·18-s − 7.09·19-s + 0.971·21-s − 15.7·22-s + 2.57·23-s − 23.0·24-s + 3.74·26-s − 0.895·27-s + 2.26·28-s + 5.12·29-s + 3.92·31-s − 23.3·32-s + 13.5·33-s + 10.8·34-s + ⋯ |
L(s) = 1 | − 1.94·2-s + 1.36·3-s + 2.76·4-s − 2.65·6-s + 0.154·7-s − 3.43·8-s + 0.874·9-s + 1.72·11-s + 3.79·12-s − 0.378·13-s − 0.300·14-s + 3.89·16-s − 0.956·17-s − 1.69·18-s − 1.62·19-s + 0.211·21-s − 3.35·22-s + 0.537·23-s − 4.70·24-s + 0.734·26-s − 0.172·27-s + 0.428·28-s + 0.951·29-s + 0.705·31-s − 4.13·32-s + 2.36·33-s + 1.85·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.383913809\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.383913809\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 151 | \( 1 - T \) |
good | 2 | \( 1 + 2.74T + 2T^{2} \) |
| 3 | \( 1 - 2.37T + 3T^{2} \) |
| 7 | \( 1 - 0.409T + 7T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 + 1.36T + 13T^{2} \) |
| 17 | \( 1 + 3.94T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 - 2.57T + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 - 3.92T + 31T^{2} \) |
| 37 | \( 1 - 2.09T + 37T^{2} \) |
| 41 | \( 1 - 0.147T + 41T^{2} \) |
| 43 | \( 1 + 1.71T + 43T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 0.776T + 59T^{2} \) |
| 61 | \( 1 - 5.83T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 1.98T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + 2.08T + 83T^{2} \) |
| 89 | \( 1 - 8.84T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−8.479159373176401419998229405782, −8.319976798869276578786505953255, −7.20642401661721683314223879033, −6.75293765201351827547821348221, −6.11780705450616099291915705087, −4.41553358305600866976570993958, −3.49055853322082862951226986888, −2.44155185191371355606515356748, −1.98737694517505200498469960723, −0.858093637301787110151015678949,
0.858093637301787110151015678949, 1.98737694517505200498469960723, 2.44155185191371355606515356748, 3.49055853322082862951226986888, 4.41553358305600866976570993958, 6.11780705450616099291915705087, 6.75293765201351827547821348221, 7.20642401661721683314223879033, 8.319976798869276578786505953255, 8.479159373176401419998229405782