| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 4·13-s + 14-s + 16-s + 6·17-s + 2·19-s − 4·26-s − 28-s + 6·29-s − 4·31-s − 32-s − 6·34-s − 2·37-s − 2·38-s − 6·41-s − 8·43-s − 12·47-s + 49-s + 4·52-s + 6·53-s + 56-s − 6·58-s + 6·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.784·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s − 0.328·37-s − 0.324·38-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.554·52-s + 0.824·53-s + 0.133·56-s − 0.787·58-s + 0.781·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.368961465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368961465\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.510181799390278872970334913519, −8.190576101673579459695630969447, −7.20568692412223659760548448204, −6.56614507384142030967141653825, −5.76860542213636603043499309352, −4.99929117585498161544137242896, −3.62404662713201803361092631083, −3.16593838634617734875675320632, −1.80953354791836556352638476370, −0.814782882121918462041385743038,
0.814782882121918462041385743038, 1.80953354791836556352638476370, 3.16593838634617734875675320632, 3.62404662713201803361092631083, 4.99929117585498161544137242896, 5.76860542213636603043499309352, 6.56614507384142030967141653825, 7.20568692412223659760548448204, 8.190576101673579459695630969447, 8.510181799390278872970334913519
