L(s) = 1 | − 0.630·2-s + 2.33·3-s − 1.60·4-s − 3.89·5-s − 1.47·6-s − 3.68·7-s + 2.27·8-s + 2.46·9-s + 2.45·10-s − 4.96·11-s − 3.74·12-s − 1.69·13-s + 2.32·14-s − 9.10·15-s + 1.77·16-s + 5.52·17-s − 1.55·18-s + 4.21·19-s + 6.24·20-s − 8.60·21-s + 3.13·22-s − 2.77·23-s + 5.31·24-s + 10.1·25-s + 1.06·26-s − 1.24·27-s + 5.90·28-s + ⋯ |
L(s) = 1 | − 0.445·2-s + 1.34·3-s − 0.801·4-s − 1.74·5-s − 0.601·6-s − 1.39·7-s + 0.803·8-s + 0.822·9-s + 0.777·10-s − 1.49·11-s − 1.08·12-s − 0.468·13-s + 0.620·14-s − 2.35·15-s + 0.442·16-s + 1.33·17-s − 0.366·18-s + 0.966·19-s + 1.39·20-s − 1.87·21-s + 0.667·22-s − 0.578·23-s + 1.08·24-s + 2.03·25-s + 0.209·26-s − 0.240·27-s + 1.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 241 | \( 1 + T \) |
good | 2 | \( 1 + 0.630T + 2T^{2} \) |
| 3 | \( 1 - 2.33T + 3T^{2} \) |
| 5 | \( 1 + 3.89T + 5T^{2} \) |
| 7 | \( 1 + 3.68T + 7T^{2} \) |
| 11 | \( 1 + 4.96T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 19 | \( 1 - 4.21T + 19T^{2} \) |
| 23 | \( 1 + 2.77T + 23T^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 + 0.199T + 31T^{2} \) |
| 37 | \( 1 - 1.79T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 5.34T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 1.32T + 53T^{2} \) |
| 59 | \( 1 - 5.78T + 59T^{2} \) |
| 61 | \( 1 + 0.0766T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 5.20T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 7.82T + 79T^{2} \) |
| 83 | \( 1 - 2.55T + 83T^{2} \) |
| 89 | \( 1 + 4.35T + 89T^{2} \) |
| 97 | \( 1 - 9.02T + 97T^{2} \) |
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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
−11.79063960355224178040041347397, −10.17434043997518723376568310917, −9.695344026124302163756742099128, −8.489925672175542386946953323801, −7.908998052440644390690202248352, −7.29897276924746874175733809289, −5.12427730936274790035134874150, −3.63381427959368206018850117794, −3.13621104260777994263964962567, 0,
3.13621104260777994263964962567, 3.63381427959368206018850117794, 5.12427730936274790035134874150, 7.29897276924746874175733809289, 7.908998052440644390690202248352, 8.489925672175542386946953323801, 9.695344026124302163756742099128, 10.17434043997518723376568310917, 11.79063960355224178040041347397