| L(s) = 1 | + 2·3-s + 4-s − 3·9-s − 3·11-s + 2·12-s + 16-s + 12·23-s − 14·27-s + 4·31-s − 6·33-s − 3·36-s + 4·37-s − 3·44-s + 24·47-s + 2·48-s − 10·49-s + 12·53-s + 64-s − 26·67-s + 24·69-s + 24·71-s − 4·81-s + 30·89-s + 12·92-s + 8·93-s + 4·97-s + 9·99-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 9-s − 0.904·11-s + 0.577·12-s + 1/4·16-s + 2.50·23-s − 2.69·27-s + 0.718·31-s − 1.04·33-s − 1/2·36-s + 0.657·37-s − 0.452·44-s + 3.50·47-s + 0.288·48-s − 1.42·49-s + 1.64·53-s + 1/8·64-s − 3.17·67-s + 2.88·69-s + 2.84·71-s − 4/9·81-s + 3.17·89-s + 1.25·92-s + 0.829·93-s + 0.406·97-s + 0.904·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.576414584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.576414584\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.942369524069980633833404820232, −8.316812877075225116243141130520, −8.003379760199122716602575955438, −7.37355750361837418705145948004, −7.29853134793764088505009214313, −6.39299024601606568879362228127, −6.05410683290230188214573869250, −5.25591445959428735491861815623, −5.20592148788576143134616479765, −4.25751151289549141315285174849, −3.54905277576412147109484999920, −2.96421704926700326951259240062, −2.65155040417235216240951299295, −2.18944587268591749591824641427, −0.870527168442070003633023384518,
0.870527168442070003633023384518, 2.18944587268591749591824641427, 2.65155040417235216240951299295, 2.96421704926700326951259240062, 3.54905277576412147109484999920, 4.25751151289549141315285174849, 5.20592148788576143134616479765, 5.25591445959428735491861815623, 6.05410683290230188214573869250, 6.39299024601606568879362228127, 7.29853134793764088505009214313, 7.37355750361837418705145948004, 8.003379760199122716602575955438, 8.316812877075225116243141130520, 8.942369524069980633833404820232
