| L(s) = 1 | − 2-s + 1.14·3-s + 4-s + 5-s − 1.14·6-s − 7-s − 8-s − 1.67·9-s − 10-s + 1.14·12-s + 3.79·13-s + 14-s + 1.14·15-s + 16-s + 6.47·17-s + 1.67·18-s + 2.67·19-s + 20-s − 1.14·21-s + 2.65·23-s − 1.14·24-s + 25-s − 3.79·26-s − 5.37·27-s − 28-s + 0.386·29-s − 1.14·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.663·3-s + 0.5·4-s + 0.447·5-s − 0.469·6-s − 0.377·7-s − 0.353·8-s − 0.559·9-s − 0.316·10-s + 0.331·12-s + 1.05·13-s + 0.267·14-s + 0.296·15-s + 0.250·16-s + 1.57·17-s + 0.395·18-s + 0.614·19-s + 0.223·20-s − 0.250·21-s + 0.553·23-s − 0.234·24-s + 0.200·25-s − 0.744·26-s − 1.03·27-s − 0.188·28-s + 0.0718·29-s − 0.209·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.115483962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.115483962\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 13 | \( 1 - 3.79T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 - 2.67T + 19T^{2} \) |
| 23 | \( 1 - 2.65T + 23T^{2} \) |
| 29 | \( 1 - 0.386T + 29T^{2} \) |
| 31 | \( 1 + 6.01T + 31T^{2} \) |
| 37 | \( 1 - 2.38T + 37T^{2} \) |
| 41 | \( 1 - 9.47T + 41T^{2} \) |
| 43 | \( 1 + 4.66T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 5.09T + 53T^{2} \) |
| 59 | \( 1 - 4.74T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 5.71T + 67T^{2} \) |
| 71 | \( 1 - 3.33T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 6.03T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 5.87T + 89T^{2} \) |
| 97 | \( 1 + 3.30T + 97T^{2} \) |
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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
−8.083696502174804214125877032759, −7.21827148342552633899744190522, −6.53992890884553092718114690588, −5.65543757028184912859262079006, −5.39580343270807929005473190096, −3.91745160870878178667563781698, −3.27862373254277757481007035569, −2.69533593598287245645977617227, −1.66719804482218482819879359091, −0.794641884309409337643998605268,
0.794641884309409337643998605268, 1.66719804482218482819879359091, 2.69533593598287245645977617227, 3.27862373254277757481007035569, 3.91745160870878178667563781698, 5.39580343270807929005473190096, 5.65543757028184912859262079006, 6.53992890884553092718114690588, 7.21827148342552633899744190522, 8.083696502174804214125877032759
