| L(s) = 1 | + 1.92·2-s − 1.39·3-s + 1.68·4-s − 2.68·6-s − 4.60·7-s − 0.600·8-s − 1.04·9-s + 1.56·11-s − 2.36·12-s − 2.60·13-s − 8.84·14-s − 4.52·16-s − 0.559·17-s − 1.99·18-s − 1.16·19-s + 6.44·21-s + 3.00·22-s + 23-s + 0.840·24-s − 4.99·26-s + 5.65·27-s − 7.77·28-s − 3.17·29-s + 10.0·31-s − 7.49·32-s − 2.19·33-s − 1.07·34-s + ⋯ |
| L(s) = 1 | + 1.35·2-s − 0.807·3-s + 0.843·4-s − 1.09·6-s − 1.74·7-s − 0.212·8-s − 0.347·9-s + 0.472·11-s − 0.681·12-s − 0.721·13-s − 2.36·14-s − 1.13·16-s − 0.135·17-s − 0.471·18-s − 0.267·19-s + 1.40·21-s + 0.641·22-s + 0.208·23-s + 0.171·24-s − 0.979·26-s + 1.08·27-s − 1.46·28-s − 0.589·29-s + 1.80·31-s − 1.32·32-s − 0.381·33-s − 0.184·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.92T + 2T^{2} \) |
| 3 | \( 1 + 1.39T + 3T^{2} \) |
| 7 | \( 1 + 4.60T + 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 13 | \( 1 + 2.60T + 13T^{2} \) |
| 17 | \( 1 + 0.559T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 + 9.32T + 47T^{2} \) |
| 53 | \( 1 - 5.54T + 53T^{2} \) |
| 59 | \( 1 - 3.84T + 59T^{2} \) |
| 61 | \( 1 + 4.29T + 61T^{2} \) |
| 67 | \( 1 + 2.60T + 67T^{2} \) |
| 71 | \( 1 - 7.89T + 71T^{2} \) |
| 73 | \( 1 + 9.90T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 7.71T + 89T^{2} \) |
| 97 | \( 1 - 2.62T + 97T^{2} \) |
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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
−10.41215448470745632287145108839, −9.608637234621926119433826044719, −8.632856558152362518318207322743, −6.87020955186298845006265579010, −6.46059058115025077899954655516, −5.63342474800200471825770591769, −4.73280117456249083193462129310, −3.57944052786861155921600525855, −2.73056148281291366437688210093, 0,
2.73056148281291366437688210093, 3.57944052786861155921600525855, 4.73280117456249083193462129310, 5.63342474800200471825770591769, 6.46059058115025077899954655516, 6.87020955186298845006265579010, 8.632856558152362518318207322743, 9.608637234621926119433826044719, 10.41215448470745632287145108839
