| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 3·11-s + 2·13-s + 14-s + 16-s − 3·17-s + 2·19-s + 20-s + 3·22-s + 25-s − 2·26-s − 28-s + 3·29-s − 31-s − 32-s + 3·34-s − 35-s + 37-s − 2·38-s − 40-s − 9·41-s + 11·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.904·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s + 0.223·20-s + 0.639·22-s + 1/5·25-s − 0.392·26-s − 0.188·28-s + 0.557·29-s − 0.179·31-s − 0.176·32-s + 0.514·34-s − 0.169·35-s + 0.164·37-s − 0.324·38-s − 0.158·40-s − 1.40·41-s + 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.208143577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.208143577\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−8.600517603595698033773674592405, −8.056328028601876196744723607128, −7.13251302424729333476135109676, −6.52308822130081203621761307573, −5.71394937860935381050566489571, −4.96694460368536127704591382321, −3.78736511669230294409797402576, −2.81048921634741052410414204103, −2.00824803168943428454242785029, −0.71673423275031954951307857151,
0.71673423275031954951307857151, 2.00824803168943428454242785029, 2.81048921634741052410414204103, 3.78736511669230294409797402576, 4.96694460368536127704591382321, 5.71394937860935381050566489571, 6.52308822130081203621761307573, 7.13251302424729333476135109676, 8.056328028601876196744723607128, 8.600517603595698033773674592405
