| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s + 7-s − 8-s + 9-s + 10-s − 11-s + 2·12-s + 4·13-s − 14-s − 2·15-s + 16-s − 18-s − 20-s + 2·21-s + 22-s − 2·24-s + 25-s − 4·26-s − 4·27-s + 28-s + 6·29-s + 2·30-s + 10·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.577·12-s + 1.10·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s − 0.223·20-s + 0.436·21-s + 0.213·22-s − 0.408·24-s + 1/5·25-s − 0.784·26-s − 0.769·27-s + 0.188·28-s + 1.11·29-s + 0.365·30-s + 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.17090933800933, −12.34411225266707, −12.11097648353336, −11.46609235422128, −11.15030762610194, −10.60602331137819, −10.14750057603946, −9.752153197541222, −9.033001981679825, −8.772611112775167, −8.310489966091865, −8.118877243587592, −7.533969689861087, −7.140092242048237, −6.472792608487897, −6.005873797982375, −5.478957438018918, −4.692765736129820, −4.177393366086915, −3.726826180684510, −3.041115830944052, −2.637013163883172, −2.233192747123590, −1.253280973241786, −1.017929589804730, 0,
1.017929589804730, 1.253280973241786, 2.233192747123590, 2.637013163883172, 3.041115830944052, 3.726826180684510, 4.177393366086915, 4.692765736129820, 5.478957438018918, 6.005873797982375, 6.472792608487897, 7.140092242048237, 7.533969689861087, 8.118877243587592, 8.310489966091865, 8.772611112775167, 9.033001981679825, 9.752153197541222, 10.14750057603946, 10.60602331137819, 11.15030762610194, 11.46609235422128, 12.11097648353336, 12.34411225266707, 13.17090933800933
