| L(s) = 1 | − 1.16e8·7-s + 6.19e10·13-s + 2.52e11·19-s − 1.90e13·25-s + 1.89e14·31-s − 7.80e14·37-s + 6.66e14·43-s + 2.14e15·49-s − 1.40e17·61-s + 2.73e17·67-s − 6.39e17·73-s − 1.96e18·79-s − 7.20e18·91-s − 9.50e18·97-s − 2.63e19·103-s − 4.39e19·109-s + ⋯ |
| L(s) = 1 | − 1.09·7-s + 1.61·13-s + 0.179·19-s − 25-s + 1.28·31-s − 0.987·37-s + 0.202·43-s + 0.188·49-s − 1.53·61-s + 1.22·67-s − 1.27·73-s − 1.84·79-s − 1.76·91-s − 1.26·97-s − 1.98·103-s − 1.93·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p^{19} T^{2} \) |
| 7 | \( 1 + 116377132 T + p^{19} T^{2} \) |
| 11 | \( 1 + p^{19} T^{2} \) |
| 13 | \( 1 - 61912545914 T + p^{19} T^{2} \) |
| 17 | \( 1 + p^{19} T^{2} \) |
| 19 | \( 1 - 252454137272 T + p^{19} T^{2} \) |
| 23 | \( 1 + p^{19} T^{2} \) |
| 29 | \( 1 + p^{19} T^{2} \) |
| 31 | \( 1 - 189181140858356 T + p^{19} T^{2} \) |
| 37 | \( 1 + 780590048220370 T + p^{19} T^{2} \) |
| 41 | \( 1 + p^{19} T^{2} \) |
| 43 | \( 1 - 666942065597816 T + p^{19} T^{2} \) |
| 47 | \( 1 + p^{19} T^{2} \) |
| 53 | \( 1 + p^{19} T^{2} \) |
| 59 | \( 1 + p^{19} T^{2} \) |
| 61 | \( 1 + 140464646991065866 T + p^{19} T^{2} \) |
| 67 | \( 1 - 273531389481429392 T + p^{19} T^{2} \) |
| 71 | \( 1 + p^{19} T^{2} \) |
| 73 | \( 1 + 639446467598381530 T + p^{19} T^{2} \) |
| 79 | \( 1 + 1962240154233379276 T + p^{19} T^{2} \) |
| 83 | \( 1 + p^{19} T^{2} \) |
| 89 | \( 1 + p^{19} T^{2} \) |
| 97 | \( 1 + 9502640950925166898 T + p^{19} T^{2} \) |
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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
−11.84787453369563278234921102863, −10.56746014657468837740071885390, −9.431773720252842499781828928212, −8.242772560706473780161502140418, −6.70673944621878840840363047115, −5.77322289985603255573326330025, −4.02452025951277129062444251124, −2.98059115288709148245702540513, −1.35106699401167180070252075005, 0,
1.35106699401167180070252075005, 2.98059115288709148245702540513, 4.02452025951277129062444251124, 5.77322289985603255573326330025, 6.70673944621878840840363047115, 8.242772560706473780161502140418, 9.431773720252842499781828928212, 10.56746014657468837740071885390, 11.84787453369563278234921102863
