| L(s) = 1 | + 2·2-s + 7.21·3-s + 4·4-s + 14.4·6-s + 19.8·7-s + 8·8-s + 24.9·9-s − 31.0·11-s + 28.8·12-s + 7.15·13-s + 39.7·14-s + 16·16-s + 40.8·17-s + 49.9·18-s + 144.·19-s + 143.·21-s − 62.1·22-s + 23·23-s + 57.6·24-s + 14.3·26-s − 14.5·27-s + 79.5·28-s − 189.·29-s + 41.6·31-s + 32·32-s − 223.·33-s + 81.6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.38·3-s + 0.5·4-s + 0.981·6-s + 1.07·7-s + 0.353·8-s + 0.925·9-s − 0.851·11-s + 0.693·12-s + 0.152·13-s + 0.759·14-s + 0.250·16-s + 0.582·17-s + 0.654·18-s + 1.75·19-s + 1.49·21-s − 0.601·22-s + 0.208·23-s + 0.490·24-s + 0.107·26-s − 0.103·27-s + 0.536·28-s − 1.21·29-s + 0.241·31-s + 0.176·32-s − 1.18·33-s + 0.411·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.644391490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.644391490\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 7.21T + 27T^{2} \) |
| 7 | \( 1 - 19.8T + 343T^{2} \) |
| 11 | \( 1 + 31.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.15T + 2.19e3T^{2} \) |
| 17 | \( 1 - 40.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 144.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 41.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 401.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 29.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 18.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 214.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 313.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 452.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 858.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 451.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 742.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 126.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 473.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 109.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 711.T + 9.12e5T^{2} \) |
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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
−9.342275802676124544011432094774, −8.452505047642268241276918292241, −7.61954678525368258673205868227, −7.41439134577891411415540797217, −5.76029784321685212899332503379, −5.09863161325096334668699848153, −4.02862620553316432377634337731, −3.13584298641386818457977846116, −2.34952604884822204348857263062, −1.25791896730636064497914942336,
1.25791896730636064497914942336, 2.34952604884822204348857263062, 3.13584298641386818457977846116, 4.02862620553316432377634337731, 5.09863161325096334668699848153, 5.76029784321685212899332503379, 7.41439134577891411415540797217, 7.61954678525368258673205868227, 8.452505047642268241276918292241, 9.342275802676124544011432094774
