| L(s) = 1 | − 1.43·2-s + 0.465·3-s + 0.0679·4-s − 0.669·6-s + 1.05·7-s + 2.77·8-s − 2.78·9-s + 1.98·11-s + 0.0316·12-s − 1.07·13-s − 1.51·14-s − 4.13·16-s − 4.36·17-s + 4.00·18-s − 1.07·19-s + 0.490·21-s − 2.85·22-s − 2.30·23-s + 1.29·24-s + 1.54·26-s − 2.69·27-s + 0.0716·28-s + 8.21·29-s + 3.23·31-s + 0.384·32-s + 0.923·33-s + 6.27·34-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 0.268·3-s + 0.0339·4-s − 0.273·6-s + 0.398·7-s + 0.982·8-s − 0.927·9-s + 0.598·11-s + 0.00913·12-s − 0.298·13-s − 0.405·14-s − 1.03·16-s − 1.05·17-s + 0.943·18-s − 0.247·19-s + 0.107·21-s − 0.608·22-s − 0.480·23-s + 0.264·24-s + 0.303·26-s − 0.518·27-s + 0.0135·28-s + 1.52·29-s + 0.580·31-s + 0.0679·32-s + 0.160·33-s + 1.07·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8461276962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8461276962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 3 | \( 1 - 0.465T + 3T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 - 1.98T + 11T^{2} \) |
| 13 | \( 1 + 1.07T + 13T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 19 | \( 1 + 1.07T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 - 8.21T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 - 0.524T + 37T^{2} \) |
| 41 | \( 1 + 9.00T + 41T^{2} \) |
| 43 | \( 1 - 5.78T + 43T^{2} \) |
| 47 | \( 1 + 7.20T + 47T^{2} \) |
| 53 | \( 1 - 6.05T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 - 9.55T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 0.286T + 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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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
−8.355297041809204935040267776386, −7.67227843088026478362545444969, −6.74115392890726238643314508591, −6.21800356743635540395703395299, −5.02787356581661220818983646625, −4.54988838675716949279863311622, −3.59494315291128683167938089620, −2.52049135371718125127039607481, −1.72715776673157948157061714135, −0.55909440013302109448871082805,
0.55909440013302109448871082805, 1.72715776673157948157061714135, 2.52049135371718125127039607481, 3.59494315291128683167938089620, 4.54988838675716949279863311622, 5.02787356581661220818983646625, 6.21800356743635540395703395299, 6.74115392890726238643314508591, 7.67227843088026478362545444969, 8.355297041809204935040267776386
