| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 3·11-s + 13-s + 14-s + 16-s + 17-s − 8·19-s − 3·22-s − 4·23-s − 26-s − 28-s + 7·29-s + 31-s − 32-s − 34-s + 4·37-s + 8·38-s + 6·41-s − 12·43-s + 3·44-s + 4·46-s − 3·47-s − 6·49-s + 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.904·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.83·19-s − 0.639·22-s − 0.834·23-s − 0.196·26-s − 0.188·28-s + 1.29·29-s + 0.179·31-s − 0.176·32-s − 0.171·34-s + 0.657·37-s + 1.29·38-s + 0.937·41-s − 1.82·43-s + 0.452·44-s + 0.589·46-s − 0.437·47-s − 6/7·49-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.943573966296973552535676921138, −6.97198353368683580025419488580, −6.35798542708256970499935133016, −6.01311084302195854219141447101, −4.72900766161738423262236972898, −4.02922264032945530209584734387, −3.13997445999593136541344926731, −2.17667557879910414717870480583, −1.26196136989823553790537167549, 0,
1.26196136989823553790537167549, 2.17667557879910414717870480583, 3.13997445999593136541344926731, 4.02922264032945530209584734387, 4.72900766161738423262236972898, 6.01311084302195854219141447101, 6.35798542708256970499935133016, 6.97198353368683580025419488580, 7.943573966296973552535676921138
