| L(s) = 1 | + 81·3-s − 3.83e3·7-s + 6.56e3·9-s − 7.63e4·11-s + 5.46e4·13-s + 1.01e5·17-s + 6.69e5·19-s − 3.10e5·21-s + 2.76e5·23-s + 5.31e5·27-s + 4.81e6·29-s + 2.34e6·31-s − 6.18e6·33-s − 8.07e6·37-s + 4.42e6·39-s + 1.49e7·41-s − 2.96e7·43-s − 1.00e7·47-s − 2.56e7·49-s + 8.18e6·51-s − 4.07e7·53-s + 5.42e7·57-s + 1.21e8·59-s + 3.38e7·61-s − 2.51e7·63-s − 2.90e8·67-s + 2.23e7·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.603·7-s + 1/3·9-s − 1.57·11-s + 0.530·13-s + 0.293·17-s + 1.17·19-s − 0.348·21-s + 0.206·23-s + 0.192·27-s + 1.26·29-s + 0.456·31-s − 0.907·33-s − 0.708·37-s + 0.306·39-s + 0.825·41-s − 1.32·43-s − 0.300·47-s − 0.635·49-s + 0.169·51-s − 0.709·53-s + 0.680·57-s + 1.30·59-s + 0.313·61-s − 0.201·63-s − 1.75·67-s + 0.118·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{4} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 548 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 76344 T + p^{9} T^{2} \) |
| 13 | \( 1 - 54670 T + p^{9} T^{2} \) |
| 17 | \( 1 - 101034 T + p^{9} T^{2} \) |
| 19 | \( 1 - 669500 T + p^{9} T^{2} \) |
| 23 | \( 1 - 276504 T + p^{9} T^{2} \) |
| 29 | \( 1 - 4815354 T + p^{9} T^{2} \) |
| 31 | \( 1 - 2348024 T + p^{9} T^{2} \) |
| 37 | \( 1 + 8072498 T + p^{9} T^{2} \) |
| 41 | \( 1 - 14935290 T + p^{9} T^{2} \) |
| 43 | \( 1 + 29629556 T + p^{9} T^{2} \) |
| 47 | \( 1 + 10067784 T + p^{9} T^{2} \) |
| 53 | \( 1 + 40751322 T + p^{9} T^{2} \) |
| 59 | \( 1 - 121173624 T + p^{9} T^{2} \) |
| 61 | \( 1 - 33880982 T + p^{9} T^{2} \) |
| 67 | \( 1 + 290012084 T + p^{9} T^{2} \) |
| 71 | \( 1 + 333711840 T + p^{9} T^{2} \) |
| 73 | \( 1 - 58019902 T + p^{9} T^{2} \) |
| 79 | \( 1 + 325929112 T + p^{9} T^{2} \) |
| 83 | \( 1 + 307125876 T + p^{9} T^{2} \) |
| 89 | \( 1 + 770779950 T + p^{9} T^{2} \) |
| 97 | \( 1 - 875079838 T + p^{9} T^{2} \) |
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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
−9.832117955736528429379471736990, −8.662996712845203825752673033854, −7.88665232884828960921892846797, −6.94187942439674416902049713418, −5.74128995331944930869266850964, −4.72651735404044127554460692015, −3.30910419676145062256983027408, −2.71343123926593112997276688564, −1.28038895039747344117805669840, 0,
1.28038895039747344117805669840, 2.71343123926593112997276688564, 3.30910419676145062256983027408, 4.72651735404044127554460692015, 5.74128995331944930869266850964, 6.94187942439674416902049713418, 7.88665232884828960921892846797, 8.662996712845203825752673033854, 9.832117955736528429379471736990
