| L(s) = 1 | − 3·2-s − 3-s + 4-s + 9·5-s + 3·6-s − 15·7-s + 21·8-s − 26·9-s − 27·10-s + 48·11-s − 12-s + 45·14-s − 9·15-s − 71·16-s + 45·17-s + 78·18-s − 6·19-s + 9·20-s + 15·21-s − 144·22-s − 162·23-s − 21·24-s − 44·25-s + 53·27-s − 15·28-s − 144·29-s + 27·30-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.192·3-s + 1/8·4-s + 0.804·5-s + 0.204·6-s − 0.809·7-s + 0.928·8-s − 0.962·9-s − 0.853·10-s + 1.31·11-s − 0.0240·12-s + 0.859·14-s − 0.154·15-s − 1.10·16-s + 0.642·17-s + 1.02·18-s − 0.0724·19-s + 0.100·20-s + 0.155·21-s − 1.39·22-s − 1.46·23-s − 0.178·24-s − 0.351·25-s + 0.377·27-s − 0.101·28-s − 0.922·29-s + 0.164·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 3 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 - 9 T + p^{3} T^{2} \) |
| 7 | \( 1 + 15 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 17 | \( 1 - 45 T + p^{3} T^{2} \) |
| 19 | \( 1 + 6 T + p^{3} T^{2} \) |
| 23 | \( 1 + 162 T + p^{3} T^{2} \) |
| 29 | \( 1 + 144 T + p^{3} T^{2} \) |
| 31 | \( 1 + 264 T + p^{3} T^{2} \) |
| 37 | \( 1 + 303 T + p^{3} T^{2} \) |
| 41 | \( 1 - 192 T + p^{3} T^{2} \) |
| 43 | \( 1 - 97 T + p^{3} T^{2} \) |
| 47 | \( 1 + 111 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 + 522 T + p^{3} T^{2} \) |
| 61 | \( 1 - 376 T + p^{3} T^{2} \) |
| 67 | \( 1 - 36 T + p^{3} T^{2} \) |
| 71 | \( 1 + 357 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1098 T + p^{3} T^{2} \) |
| 79 | \( 1 + 830 T + p^{3} T^{2} \) |
| 83 | \( 1 - 438 T + p^{3} T^{2} \) |
| 89 | \( 1 - 438 T + p^{3} T^{2} \) |
| 97 | \( 1 - 852 T + p^{3} T^{2} \) |
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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
−11.62913383846420608027788393970, −10.52894250256668186065890851512, −9.514275288248203133143482532330, −9.128369676463391838822139314324, −7.87032157871618943125775487934, −6.52446706613692604068320335974, −5.57505833111046824806377405049, −3.72416006473091300457387639391, −1.76609562981741149330153595510, 0,
1.76609562981741149330153595510, 3.72416006473091300457387639391, 5.57505833111046824806377405049, 6.52446706613692604068320335974, 7.87032157871618943125775487934, 9.128369676463391838822139314324, 9.514275288248203133143482532330, 10.52894250256668186065890851512, 11.62913383846420608027788393970
