| L(s) = 1 | − 544·2-s − 5.90e4·3-s − 1.80e6·4-s + 3.21e7·6-s − 1.27e9·7-s + 2.12e9·8-s + 3.48e9·9-s − 7.75e10·11-s + 1.06e11·12-s + 4.34e11·13-s + 6.95e11·14-s + 2.62e12·16-s − 1.28e13·17-s − 1.89e12·18-s − 2.86e13·19-s + 7.54e13·21-s + 4.22e13·22-s − 2.24e14·23-s − 1.25e14·24-s − 2.36e14·26-s − 2.05e14·27-s + 2.30e15·28-s − 5.16e13·29-s + 8.92e15·31-s − 5.87e15·32-s + 4.58e15·33-s + 6.98e15·34-s + ⋯ |
| L(s) = 1 | − 0.375·2-s − 0.577·3-s − 0.858·4-s + 0.216·6-s − 1.70·7-s + 0.698·8-s + 1/3·9-s − 0.901·11-s + 0.495·12-s + 0.873·13-s + 0.642·14-s + 0.596·16-s − 1.54·17-s − 0.125·18-s − 1.07·19-s + 0.987·21-s + 0.338·22-s − 1.12·23-s − 0.403·24-s − 0.328·26-s − 0.192·27-s + 1.46·28-s − 0.0228·29-s + 1.95·31-s − 0.922·32-s + 0.520·33-s + 0.580·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{10} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 17 p^{5} T + p^{21} T^{2} \) |
| 7 | \( 1 + 182528340 p T + p^{21} T^{2} \) |
| 11 | \( 1 + 77585921744 T + p^{21} T^{2} \) |
| 13 | \( 1 - 33393146054 p T + p^{21} T^{2} \) |
| 17 | \( 1 + 12848917115782 T + p^{21} T^{2} \) |
| 19 | \( 1 + 1505539797884 p T + p^{21} T^{2} \) |
| 23 | \( 1 + 224022192208080 T + p^{21} T^{2} \) |
| 29 | \( 1 + 51676030833142 T + p^{21} T^{2} \) |
| 31 | \( 1 - 8921108838285000 T + p^{21} T^{2} \) |
| 37 | \( 1 + 43977154002495890 T + p^{21} T^{2} \) |
| 41 | \( 1 - 58168090830044570 T + p^{21} T^{2} \) |
| 43 | \( 1 - 161437862491900676 T + p^{21} T^{2} \) |
| 47 | \( 1 - 160064774442316592 T + p^{21} T^{2} \) |
| 53 | \( 1 + 2299527285858152170 T + p^{21} T^{2} \) |
| 59 | \( 1 - 5154256088898000016 T + p^{21} T^{2} \) |
| 61 | \( 1 - 1251686105775241798 T + p^{21} T^{2} \) |
| 67 | \( 1 - 5407785329527117188 T + p^{21} T^{2} \) |
| 71 | \( 1 + 11043230850518282368 T + p^{21} T^{2} \) |
| 73 | \( 1 - 37701191520217147550 T + p^{21} T^{2} \) |
| 79 | \( 1 - 63155369968366862760 T + p^{21} T^{2} \) |
| 83 | \( 1 - \)\(14\!\cdots\!28\)\( T + p^{21} T^{2} \) |
| 89 | \( 1 - \)\(13\!\cdots\!14\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 - \)\(32\!\cdots\!18\)\( T + p^{21} T^{2} \) |
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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
−10.12430443986936619032610658007, −9.132314654591199964475582672591, −8.176246624904199904869565162418, −6.67816686097845109077495392357, −5.95497589827591452556916521019, −4.58463518764228007326726305716, −3.67175391918077440918982285290, −2.29013981297871465169854655600, −0.65839666090119083342998733098, 0,
0.65839666090119083342998733098, 2.29013981297871465169854655600, 3.67175391918077440918982285290, 4.58463518764228007326726305716, 5.95497589827591452556916521019, 6.67816686097845109077495392357, 8.176246624904199904869565162418, 9.132314654591199964475582672591, 10.12430443986936619032610658007
