| L(s) = 1 | − 2.63·2-s + 4.91·4-s + 0.574·7-s − 7.67·8-s − 2.57·11-s + 0.468·13-s − 1.51·14-s + 10.3·16-s − 4.08·17-s + 19-s + 6.77·22-s + 1.51·23-s − 1.23·26-s + 2.82·28-s + 4.08·29-s − 9.92·31-s − 11.8·32-s + 10.7·34-s + 8.30·37-s − 2.63·38-s + 1.83·41-s + 0.574·43-s − 12.6·44-s − 3.97·46-s + 7.09·47-s − 6.66·49-s + 2.30·52-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.45·4-s + 0.217·7-s − 2.71·8-s − 0.776·11-s + 0.129·13-s − 0.403·14-s + 2.58·16-s − 0.991·17-s + 0.229·19-s + 1.44·22-s + 0.315·23-s − 0.241·26-s + 0.534·28-s + 0.758·29-s − 1.78·31-s − 2.09·32-s + 1.84·34-s + 1.36·37-s − 0.426·38-s + 0.286·41-s + 0.0876·43-s − 1.90·44-s − 0.586·46-s + 1.03·47-s − 0.952·49-s + 0.319·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 7 | \( 1 - 0.574T + 7T^{2} \) |
| 11 | \( 1 + 2.57T + 11T^{2} \) |
| 13 | \( 1 - 0.468T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 - 4.08T + 29T^{2} \) |
| 31 | \( 1 + 9.92T + 31T^{2} \) |
| 37 | \( 1 - 8.30T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 43 | \( 1 - 0.574T + 43T^{2} \) |
| 47 | \( 1 - 7.09T + 47T^{2} \) |
| 53 | \( 1 - 4.30T + 53T^{2} \) |
| 59 | \( 1 - 2.68T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 6.68T + 79T^{2} \) |
| 83 | \( 1 + 6.66T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.299892100198900686853332167273, −7.37224171912663930321246205673, −7.01822437908197716563738123955, −6.08666412585599110291066466628, −5.31087910730102560818751912892, −4.12045586024953600311788464283, −2.84235017634127722128000906342, −2.20819034675622927177221193097, −1.14881409513388034603261793924, 0,
1.14881409513388034603261793924, 2.20819034675622927177221193097, 2.84235017634127722128000906342, 4.12045586024953600311788464283, 5.31087910730102560818751912892, 6.08666412585599110291066466628, 7.01822437908197716563738123955, 7.37224171912663930321246205673, 8.299892100198900686853332167273
