| L(s) = 1 | + 2-s + 4-s − 1.23·7-s + 8-s + 4.47·11-s + 13-s − 1.23·14-s + 16-s − 2.76·17-s − 7.23·19-s + 4.47·22-s − 7.23·23-s + 26-s − 1.23·28-s − 9.70·29-s − 4·31-s + 32-s − 2.76·34-s + 6.94·37-s − 7.23·38-s − 12.4·41-s + 6.47·43-s + 4.47·44-s − 7.23·46-s + 4.94·47-s − 5.47·49-s + 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.467·7-s + 0.353·8-s + 1.34·11-s + 0.277·13-s − 0.330·14-s + 0.250·16-s − 0.670·17-s − 1.66·19-s + 0.953·22-s − 1.50·23-s + 0.196·26-s − 0.233·28-s − 1.80·29-s − 0.718·31-s + 0.176·32-s − 0.474·34-s + 1.14·37-s − 1.17·38-s − 1.94·41-s + 0.986·43-s + 0.674·44-s − 1.06·46-s + 0.721·47-s − 0.781·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 7.23T + 23T^{2} \) |
| 29 | \( 1 + 9.70T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 4.94T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 0.472T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 8.47T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−7.61383602082054390349132229286, −6.79015900919429221608068355790, −6.25026473738805906660265384068, −5.80670627044348311115959358704, −4.66019785843464519839118536704, −3.95680711280824626523750376584, −3.58310223646203193591377475995, −2.29395518141471516794836985667, −1.64697555635037653880263001564, 0,
1.64697555635037653880263001564, 2.29395518141471516794836985667, 3.58310223646203193591377475995, 3.95680711280824626523750376584, 4.66019785843464519839118536704, 5.80670627044348311115959358704, 6.25026473738805906660265384068, 6.79015900919429221608068355790, 7.61383602082054390349132229286
