| L(s) = 1 | − 2-s + 2.21·3-s + 4-s − 2.21·6-s + 3.90·7-s − 8-s + 1.90·9-s − 1.06·11-s + 2.21·12-s − 3.90·14-s + 16-s + 1.31·17-s − 1.90·18-s + 6.59·19-s + 8.64·21-s + 1.06·22-s + 2.14·23-s − 2.21·24-s − 2.42·27-s + 3.90·28-s − 9.05·29-s + 6.92·31-s − 32-s − 2.36·33-s − 1.31·34-s + 1.90·36-s − 5.02·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.27·3-s + 0.5·4-s − 0.903·6-s + 1.47·7-s − 0.353·8-s + 0.634·9-s − 0.321·11-s + 0.639·12-s − 1.04·14-s + 0.250·16-s + 0.317·17-s − 0.448·18-s + 1.51·19-s + 1.88·21-s + 0.227·22-s + 0.447·23-s − 0.451·24-s − 0.467·27-s + 0.737·28-s − 1.68·29-s + 1.24·31-s − 0.176·32-s − 0.411·33-s − 0.224·34-s + 0.317·36-s − 0.825·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.106392366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.106392366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 7 | \( 1 - 3.90T + 7T^{2} \) |
| 11 | \( 1 + 1.06T + 11T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 - 6.59T + 19T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 + 9.05T + 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 + 5.02T + 37T^{2} \) |
| 41 | \( 1 - 5.95T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 0.0967T + 47T^{2} \) |
| 53 | \( 1 + 1.49T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 7.88T + 61T^{2} \) |
| 67 | \( 1 - 8.42T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 4.30T + 79T^{2} \) |
| 83 | \( 1 + 9.69T + 83T^{2} \) |
| 89 | \( 1 + 4.52T + 89T^{2} \) |
| 97 | \( 1 - 4.26T + 97T^{2} \) |
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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
−7.940444619452474428060650239737, −7.53196720452950221655550414345, −6.77064472304152220740816050498, −5.53975427309502727244197572570, −5.13928329718459054665094260757, −4.05756252718051871435919484196, −3.29529743443471423943595909338, −2.49936113475998148990189835117, −1.80515593654174152997646255339, −0.949272838457246876165405949611,
0.949272838457246876165405949611, 1.80515593654174152997646255339, 2.49936113475998148990189835117, 3.29529743443471423943595909338, 4.05756252718051871435919484196, 5.13928329718459054665094260757, 5.53975427309502727244197572570, 6.77064472304152220740816050498, 7.53196720452950221655550414345, 7.940444619452474428060650239737
