| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s − 7-s + 8-s + 9-s + 2·12-s + 4·13-s − 14-s + 16-s − 6·17-s + 18-s + 2·19-s − 2·21-s + 2·24-s + 4·26-s − 4·27-s − 28-s − 6·29-s − 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 2·38-s + 8·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.458·19-s − 0.436·21-s + 0.408·24-s + 0.784·26-s − 0.769·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.658249180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.658249180\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 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
−11.46841286601878794911833201781, −10.74873683533699837976941407431, −9.395997320196623402140164716317, −8.758164081717971221129539824522, −7.71895664384636679391588711627, −6.66965130292041403123447657278, −5.59857578974307577358363282055, −4.11500666640753372808858013263, −3.28327332646103144207952017478, −2.07204529193989626181377807359,
2.07204529193989626181377807359, 3.28327332646103144207952017478, 4.11500666640753372808858013263, 5.59857578974307577358363282055, 6.66965130292041403123447657278, 7.71895664384636679391588711627, 8.758164081717971221129539824522, 9.395997320196623402140164716317, 10.74873683533699837976941407431, 11.46841286601878794911833201781
