| L(s) = 1 | − 4·11-s + 10·19-s + 16·29-s + 14·31-s − 12·41-s − 2·49-s + 8·59-s + 26·61-s + 12·71-s − 18·79-s − 12·89-s − 24·101-s − 34·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 2.29·19-s + 2.97·29-s + 2.51·31-s − 1.87·41-s − 2/7·49-s + 1.04·59-s + 3.32·61-s + 1.42·71-s − 2.02·79-s − 1.27·89-s − 2.38·101-s − 3.25·109-s − 0.909·121-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 + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.791934953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.791934953\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 123 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.457513604577591734207474731755, −7.912628774792344586356678604113, −7.888779604446513700235520831823, −7.27377081650651491099368540753, −6.71930137280208014404550719121, −6.64839939934742245942584932295, −6.48498415433447582404446566068, −5.63898729066692991743344834641, −5.34373350150100179052004926480, −5.07645861724610998857834536149, −4.98451315301008518332294637764, −4.22034593240676834943524441166, −4.00607476183968351693958705982, −3.38695235142391045594797925299, −2.80163996766485035779918020571, −2.73570912604652435023129315319, −2.44384031563773400378362663838, −1.28163865748179823440418228252, −1.21879466829391126529469352461, −0.49394191554712047749289178925,
0.49394191554712047749289178925, 1.21879466829391126529469352461, 1.28163865748179823440418228252, 2.44384031563773400378362663838, 2.73570912604652435023129315319, 2.80163996766485035779918020571, 3.38695235142391045594797925299, 4.00607476183968351693958705982, 4.22034593240676834943524441166, 4.98451315301008518332294637764, 5.07645861724610998857834536149, 5.34373350150100179052004926480, 5.63898729066692991743344834641, 6.48498415433447582404446566068, 6.64839939934742245942584932295, 6.71930137280208014404550719121, 7.27377081650651491099368540753, 7.888779604446513700235520831823, 7.912628774792344586356678604113, 8.457513604577591734207474731755
