| L(s) = 1 | + 3·4-s + 10·11-s + 5·16-s + 4·19-s − 20·29-s + 12·31-s + 20·41-s + 30·44-s + 14·49-s − 10·59-s − 22·61-s + 3·64-s − 10·71-s + 12·76-s − 24·79-s − 4·109-s − 60·116-s + 53·121-s + 36·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 3.01·11-s + 5/4·16-s + 0.917·19-s − 3.71·29-s + 2.15·31-s + 3.12·41-s + 4.52·44-s + 2·49-s − 1.30·59-s − 2.81·61-s + 3/8·64-s − 1.18·71-s + 1.37·76-s − 2.70·79-s − 0.383·109-s − 5.57·116-s + 4.81·121-s + 3.23·124-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 + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.689680886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.689680886\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 69 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
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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
−10.76194932175328185197456336510, −10.47795272927220041038251234258, −9.692834763594593047562485242102, −9.347095673279079830702906899385, −9.142098032894666269382912181226, −8.780244305684184706248427874218, −7.81957016131671945697141976445, −7.52497004695439709140936341602, −7.21561980281275487324901697518, −6.81091316154194851102349603580, −6.05299679203962949775502701059, −6.05066771468867602560303849696, −5.68037805462495679283598631365, −4.49277104795619184155909624436, −4.18442990003152948110809491113, −3.66298185132715669005156177294, −3.02421036653008112002297873733, −2.39790218085434868911691391704, −1.50113226444934803839057470667, −1.24903375147205186177847545808,
1.24903375147205186177847545808, 1.50113226444934803839057470667, 2.39790218085434868911691391704, 3.02421036653008112002297873733, 3.66298185132715669005156177294, 4.18442990003152948110809491113, 4.49277104795619184155909624436, 5.68037805462495679283598631365, 6.05066771468867602560303849696, 6.05299679203962949775502701059, 6.81091316154194851102349603580, 7.21561980281275487324901697518, 7.52497004695439709140936341602, 7.81957016131671945697141976445, 8.780244305684184706248427874218, 9.142098032894666269382912181226, 9.347095673279079830702906899385, 9.692834763594593047562485242102, 10.47795272927220041038251234258, 10.76194932175328185197456336510
