| L(s) = 1 | + 76·2-s − 729·3-s − 2.41e3·4-s − 5.54e4·6-s + 2.24e5·7-s − 8.06e5·8-s + 5.31e5·9-s + 2.31e6·11-s + 1.76e6·12-s − 1.05e7·13-s + 1.70e7·14-s − 4.14e7·16-s + 1.86e8·17-s + 4.03e7·18-s − 2.90e8·19-s − 1.63e8·21-s + 1.75e8·22-s + 8.66e8·23-s + 5.87e8·24-s − 8.00e8·26-s − 3.87e8·27-s − 5.41e8·28-s − 1.56e9·29-s + 1.20e9·31-s + 3.45e9·32-s − 1.68e9·33-s + 1.41e10·34-s + ⋯ |
| L(s) = 1 | + 0.839·2-s − 0.577·3-s − 0.294·4-s − 0.484·6-s + 0.720·7-s − 1.08·8-s + 1/3·9-s + 0.393·11-s + 0.170·12-s − 0.605·13-s + 0.604·14-s − 0.618·16-s + 1.87·17-s + 0.279·18-s − 1.41·19-s − 0.415·21-s + 0.330·22-s + 1.22·23-s + 0.627·24-s − 0.508·26-s − 0.192·27-s − 0.212·28-s − 0.489·29-s + 0.242·31-s + 0.568·32-s − 0.227·33-s + 1.57·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{6} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 19 p^{2} T + p^{13} T^{2} \) |
| 7 | \( 1 - 224160 T + p^{13} T^{2} \) |
| 11 | \( 1 - 2313836 T + p^{13} T^{2} \) |
| 13 | \( 1 + 10537318 T + p^{13} T^{2} \) |
| 17 | \( 1 - 186660598 T + p^{13} T^{2} \) |
| 19 | \( 1 + 290440676 T + p^{13} T^{2} \) |
| 23 | \( 1 - 866818560 T + p^{13} T^{2} \) |
| 29 | \( 1 + 1566981362 T + p^{13} T^{2} \) |
| 31 | \( 1 - 1200623400 T + p^{13} T^{2} \) |
| 37 | \( 1 - 12182249410 T + p^{13} T^{2} \) |
| 41 | \( 1 + 29167834310 T + p^{13} T^{2} \) |
| 43 | \( 1 + 49361767564 T + p^{13} T^{2} \) |
| 47 | \( 1 - 11671527832 T + p^{13} T^{2} \) |
| 53 | \( 1 + 100929409430 T + p^{13} T^{2} \) |
| 59 | \( 1 + 265189749604 T + p^{13} T^{2} \) |
| 61 | \( 1 + 566433594722 T + p^{13} T^{2} \) |
| 67 | \( 1 + 1441180693572 T + p^{13} T^{2} \) |
| 71 | \( 1 + 502944753848 T + p^{13} T^{2} \) |
| 73 | \( 1 + 1574910852730 T + p^{13} T^{2} \) |
| 79 | \( 1 - 338387056680 T + p^{13} T^{2} \) |
| 83 | \( 1 - 4771809968748 T + p^{13} T^{2} \) |
| 89 | \( 1 - 2746483865994 T + p^{13} T^{2} \) |
| 97 | \( 1 + 1979074481282 T + p^{13} T^{2} \) |
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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
−11.67636013085509399716977810706, −10.39322074346291661073038055888, −9.147253497286670103435670001281, −7.82151689774848751670073424215, −6.35004934860158981691447474159, −5.24273270672847895537583999549, −4.46487816230697507245235320428, −3.15178827925286254598281272349, −1.40652166425563595600430686911, 0,
1.40652166425563595600430686911, 3.15178827925286254598281272349, 4.46487816230697507245235320428, 5.24273270672847895537583999549, 6.35004934860158981691447474159, 7.82151689774848751670073424215, 9.147253497286670103435670001281, 10.39322074346291661073038055888, 11.67636013085509399716977810706
