| L(s) = 1 | + 3·4-s + 2·11-s + 5·16-s + 16·19-s + 2·29-s + 6·31-s + 4·41-s + 6·44-s + 5·49-s + 10·59-s + 2·61-s + 3·64-s − 32·71-s + 48·76-s − 24·79-s − 18·101-s + 20·109-s + 6·116-s − 19·121-s + 18·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 0.603·11-s + 5/4·16-s + 3.67·19-s + 0.371·29-s + 1.07·31-s + 0.624·41-s + 0.904·44-s + 5/7·49-s + 1.30·59-s + 0.256·61-s + 3/8·64-s − 3.79·71-s + 5.50·76-s − 2.70·79-s − 1.79·101-s + 1.91·109-s + 0.557·116-s − 1.72·121-s + 1.61·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8555625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.599433342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.599433342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 157 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.923448782788141767812499077872, −8.536075188297016073188684519429, −8.139830759317546178029753897431, −7.52388132689430405389929909535, −7.39234293722107268220631783082, −7.20639364535190907230784114761, −6.78764447935402384712102073594, −6.27608067808681876030254641108, −5.94751400576268089224502979778, −5.47958233966106311363163820785, −5.36364189423343105860045754700, −4.63755240687546423405564849878, −4.22787468512535553114394299575, −3.66455015448418708207104958393, −3.11525308982917661605294951285, −2.83034311294789895943277843506, −2.59271083236778454334353384430, −1.61841926705566144331807660126, −1.34254585714859250928241258766, −0.802292087908999708843012353610,
0.802292087908999708843012353610, 1.34254585714859250928241258766, 1.61841926705566144331807660126, 2.59271083236778454334353384430, 2.83034311294789895943277843506, 3.11525308982917661605294951285, 3.66455015448418708207104958393, 4.22787468512535553114394299575, 4.63755240687546423405564849878, 5.36364189423343105860045754700, 5.47958233966106311363163820785, 5.94751400576268089224502979778, 6.27608067808681876030254641108, 6.78764447935402384712102073594, 7.20639364535190907230784114761, 7.39234293722107268220631783082, 7.52388132689430405389929909535, 8.139830759317546178029753897431, 8.536075188297016073188684519429, 8.923448782788141767812499077872
