| L(s) = 1 | − 3-s − 2·4-s + 3·7-s − 6·11-s + 2·12-s − 7·13-s + 6·19-s − 3·21-s − 6·23-s + 27-s − 6·28-s − 6·29-s + 6·33-s + 7·39-s − 12·41-s − 43-s + 12·44-s − 49-s + 14·52-s − 24·53-s − 6·57-s − 6·59-s − 61-s + 8·64-s − 15·67-s + 6·69-s − 18·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1.13·7-s − 1.80·11-s + 0.577·12-s − 1.94·13-s + 1.37·19-s − 0.654·21-s − 1.25·23-s + 0.192·27-s − 1.13·28-s − 1.11·29-s + 1.04·33-s + 1.12·39-s − 1.87·41-s − 0.152·43-s + 1.80·44-s − 1/7·49-s + 1.94·52-s − 3.29·53-s − 0.794·57-s − 0.781·59-s − 0.128·61-s + 64-s − 1.83·67-s + 0.722·69-s − 2.13·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 18 T + 179 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.670613633660325487730464257980, −9.651715615506368687536095045272, −8.939055122074977073629508138444, −8.584250861254240999058392445136, −7.911030600551024386461923548373, −7.75599880232631505058479938482, −7.50064747390494244364191570336, −7.02286014279295953272144916573, −6.18808933732839416372932720253, −5.75750726649752517900508221771, −5.17307383378504087078451922997, −5.08615404826134296847720932980, −4.49952413052877564035388332568, −4.48956763572689170067062061150, −3.21632049637395201077392991657, −3.04759202969689392374417843759, −2.06555189694570825403784157666, −1.62653516659400084362747731031, 0, 0,
1.62653516659400084362747731031, 2.06555189694570825403784157666, 3.04759202969689392374417843759, 3.21632049637395201077392991657, 4.48956763572689170067062061150, 4.49952413052877564035388332568, 5.08615404826134296847720932980, 5.17307383378504087078451922997, 5.75750726649752517900508221771, 6.18808933732839416372932720253, 7.02286014279295953272144916573, 7.50064747390494244364191570336, 7.75599880232631505058479938482, 7.911030600551024386461923548373, 8.584250861254240999058392445136, 8.939055122074977073629508138444, 9.651715615506368687536095045272, 9.670613633660325487730464257980
