| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s − 11-s + 12-s + 13-s + 14-s − 15-s + 16-s − 6·17-s + 18-s − 4·19-s − 20-s + 21-s − 22-s + 24-s + 25-s + 26-s + 27-s + 28-s + 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.29781039002966, −14.92816354003967, −14.31248220400807, −13.82555259799804, −13.31126736894267, −12.85925317515131, −12.32743166579732, −11.74268720040111, −11.09098446627026, −10.74120317296329, −10.20491228030318, −9.334432850385673, −8.725467108046911, −8.360482795336107, −7.724129368863629, −7.070183663317457, −6.600167058888214, −5.938548424565094, −5.169428882661342, −4.416090321988619, −4.216369937366734, −3.404736032999961, −2.624002727486368, −2.148676547671218, −1.247403740066371, 0,
1.247403740066371, 2.148676547671218, 2.624002727486368, 3.404736032999961, 4.216369937366734, 4.416090321988619, 5.169428882661342, 5.938548424565094, 6.600167058888214, 7.070183663317457, 7.724129368863629, 8.360482795336107, 8.725467108046911, 9.334432850385673, 10.20491228030318, 10.74120317296329, 11.09098446627026, 11.74268720040111, 12.32743166579732, 12.85925317515131, 13.31126736894267, 13.82555259799804, 14.31248220400807, 14.92816354003967, 15.29781039002966
