| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s + 3·11-s + 5·13-s + 16-s − 3·17-s + 5·19-s + 3·20-s − 3·22-s + 3·23-s + 4·25-s − 5·26-s + 3·29-s − 4·31-s − 32-s + 3·34-s − 7·37-s − 5·38-s − 3·40-s + 9·41-s + 11·43-s + 3·44-s − 3·46-s − 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s + 0.904·11-s + 1.38·13-s + 1/4·16-s − 0.727·17-s + 1.14·19-s + 0.670·20-s − 0.639·22-s + 0.625·23-s + 4/5·25-s − 0.980·26-s + 0.557·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s − 1.15·37-s − 0.811·38-s − 0.474·40-s + 1.40·41-s + 1.67·43-s + 0.452·44-s − 0.442·46-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.480186275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480186275\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.896002824468772867124887944069, −7.07202718802802904935959006447, −6.46875631266383359362488705130, −5.91268239024181105516050122771, −5.32856756153549633712530519017, −4.22867621208890304786890304181, −3.36102455812602222403938447917, −2.46283040555344333262666388733, −1.56446423333794683476342273528, −0.968458189328790095257951357426,
0.968458189328790095257951357426, 1.56446423333794683476342273528, 2.46283040555344333262666388733, 3.36102455812602222403938447917, 4.22867621208890304786890304181, 5.32856756153549633712530519017, 5.91268239024181105516050122771, 6.46875631266383359362488705130, 7.07202718802802904935959006447, 7.896002824468772867124887944069
