L(s) = 1 | − 2·5-s + 7-s − 3·9-s + 4·11-s + 2·13-s + 6·17-s − 8·19-s − 25-s + 6·29-s + 8·31-s − 2·35-s + 2·37-s + 2·41-s + 4·43-s + 6·45-s − 8·47-s + 49-s − 6·53-s − 8·55-s + 6·61-s − 3·63-s − 4·65-s + 4·67-s − 8·71-s + 10·73-s + 4·77-s − 16·79-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s − 9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.338·35-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 0.768·61-s − 0.377·63-s − 0.496·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s + 0.455·77-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.915172746\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.915172746\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.04519744846792, −14.55053095698090, −14.25960759934653, −13.77412719300944, −12.86654681661842, −12.42712117718833, −11.81119051885470, −11.48748261303917, −11.09828597118221, −10.27998180275509, −9.870376718788180, −8.935845240508248, −8.549237907012132, −8.125633930135934, −7.643987437516891, −6.742936317462425, −6.212688749657977, −5.839972046757178, −4.809592889761468, −4.355235053784653, −3.685539755949762, −3.145260822601394, −2.306639290429212, −1.330761570360522, −0.5771568425087230,
0.5771568425087230, 1.330761570360522, 2.306639290429212, 3.145260822601394, 3.685539755949762, 4.355235053784653, 4.809592889761468, 5.839972046757178, 6.212688749657977, 6.742936317462425, 7.643987437516891, 8.125633930135934, 8.549237907012132, 8.935845240508248, 9.870376718788180, 10.27998180275509, 11.09828597118221, 11.48748261303917, 11.81119051885470, 12.42712117718833, 12.86654681661842, 13.77412719300944, 14.25960759934653, 14.55053095698090, 15.04519744846792