| L(s) = 1 | − 1.33·2-s − 1.71·3-s + 0.770·4-s + 1.79·5-s + 2.28·6-s − 1.54·7-s + 0.305·8-s + 1.95·9-s − 2.38·10-s + 1.44·11-s − 1.32·12-s + 2.05·14-s − 3.08·15-s − 1.17·16-s − 2.60·18-s + 1.38·20-s + 2.65·21-s − 1.91·22-s − 0.524·24-s + 2.21·25-s − 1.64·27-s − 1.18·28-s − 1.85·29-s + 4.10·30-s + 1.26·32-s − 2.47·33-s − 2.76·35-s + ⋯ |
| L(s) = 1 | − 1.33·2-s − 1.71·3-s + 0.770·4-s + 1.79·5-s + 2.28·6-s − 1.54·7-s + 0.305·8-s + 1.95·9-s − 2.38·10-s + 1.44·11-s − 1.32·12-s + 2.05·14-s − 3.08·15-s − 1.17·16-s − 2.60·18-s + 1.38·20-s + 2.65·21-s − 1.91·22-s − 0.524·24-s + 2.21·25-s − 1.64·27-s − 1.18·28-s − 1.85·29-s + 4.10·30-s + 1.26·32-s − 2.47·33-s − 2.76·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3704717765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3704717765\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1151 | \( 1 - T \) |
| good | 2 | \( 1 + 1.33T + T^{2} \) |
| 3 | \( 1 + 1.71T + T^{2} \) |
| 5 | \( 1 - 1.79T + T^{2} \) |
| 7 | \( 1 + 1.54T + T^{2} \) |
| 11 | \( 1 - 1.44T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.85T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.818T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.63T + T^{2} \) |
| 47 | \( 1 + 0.529T + T^{2} \) |
| 53 | \( 1 - 1.90T + T^{2} \) |
| 59 | \( 1 - 1.21T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.97T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 0.380T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.833472802451220131463753647882, −9.498858880055757266376967668830, −8.891193185868620661426994848962, −7.06706937960541225296991051352, −6.71612223491967000992715075243, −5.98116463642020803505684681953, −5.37207477394695775666981947384, −3.94323197572950302455677189546, −2.05648194940788566989001570881, −0.936941154694994616074170576293,
0.936941154694994616074170576293, 2.05648194940788566989001570881, 3.94323197572950302455677189546, 5.37207477394695775666981947384, 5.98116463642020803505684681953, 6.71612223491967000992715075243, 7.06706937960541225296991051352, 8.891193185868620661426994848962, 9.498858880055757266376967668830, 9.833472802451220131463753647882
