| L(s) = 1 | + 3-s + 7-s − 9-s + 2·11-s − 3·13-s − 7·17-s − 6·19-s + 21-s + 2·23-s + 7·29-s + 2·31-s + 2·33-s + 2·37-s − 3·39-s − 12·41-s + 3·43-s + 12·47-s − 9·49-s − 7·51-s − 10·53-s − 6·57-s − 8·59-s − 12·61-s − 63-s − 7·67-s + 2·69-s + 6·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s − 1/3·9-s + 0.603·11-s − 0.832·13-s − 1.69·17-s − 1.37·19-s + 0.218·21-s + 0.417·23-s + 1.29·29-s + 0.359·31-s + 0.348·33-s + 0.328·37-s − 0.480·39-s − 1.87·41-s + 0.457·43-s + 1.75·47-s − 9/7·49-s − 0.980·51-s − 1.37·53-s − 0.794·57-s − 1.04·59-s − 1.53·61-s − 0.125·63-s − 0.855·67-s + 0.240·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 66 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_4$ | \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 142 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 134 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 133 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T - 48 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 153 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 284 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.52866226868380152472541426917, −7.27092848330043864981750836766, −6.76939703877494470679295189402, −6.52687848430293233086844458956, −6.27655815112257022678716289887, −6.09322956478084460671637428084, −5.27977461887273242641463275075, −5.13527267941128173740538555806, −4.62017918192067456788682050577, −4.42834197617466056051188592375, −4.17606689321520368742281394589, −3.67055856411685739705624743794, −2.96345814534983309324567179457, −2.95535430058418892422802130602, −2.39848279691970269184132344148, −2.09085422950409717680777800459, −1.52993192763127064030013848171, −1.13576512018787292269590161908, 0, 0,
1.13576512018787292269590161908, 1.52993192763127064030013848171, 2.09085422950409717680777800459, 2.39848279691970269184132344148, 2.95535430058418892422802130602, 2.96345814534983309324567179457, 3.67055856411685739705624743794, 4.17606689321520368742281394589, 4.42834197617466056051188592375, 4.62017918192067456788682050577, 5.13527267941128173740538555806, 5.27977461887273242641463275075, 6.09322956478084460671637428084, 6.27655815112257022678716289887, 6.52687848430293233086844458956, 6.76939703877494470679295189402, 7.27092848330043864981750836766, 7.52866226868380152472541426917
