| L(s) = 1 | + 2·7-s − 4·9-s + 12·17-s − 10·25-s − 20·31-s − 20·41-s + 4·47-s + 3·49-s − 8·63-s − 24·71-s + 4·73-s + 8·79-s + 7·81-s − 4·89-s − 4·97-s + 12·103-s + 24·113-s + 24·119-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 4/3·9-s + 2.91·17-s − 2·25-s − 3.59·31-s − 3.12·41-s + 0.583·47-s + 3/7·49-s − 1.00·63-s − 2.84·71-s + 0.468·73-s + 0.900·79-s + 7/9·81-s − 0.423·89-s − 0.406·97-s + 1.18·103-s + 2.25·113-s + 2.20·119-s − 1.27·121-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 − 3.88·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51380224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51380224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−7.66775686795789806282976278921, −7.46473132756715875357590270316, −7.30671405304916208092400729211, −6.80882354276778293913162940303, −6.02494145781311797519071895620, −5.97728761123171735345903077564, −5.58427485792964587314152582088, −5.46363929357380701737020358050, −4.92092142676527129987847930432, −4.82663288990400200190820024985, −3.83208359418711170321037230257, −3.71962391070663824146360538192, −3.43957009147269527585473042151, −3.11678883485340798126744981664, −2.39340479472957169323856885696, −2.03573406855574972793299962833, −1.47176475033410517924645929793, −1.24047120954098137707754470380, 0, 0,
1.24047120954098137707754470380, 1.47176475033410517924645929793, 2.03573406855574972793299962833, 2.39340479472957169323856885696, 3.11678883485340798126744981664, 3.43957009147269527585473042151, 3.71962391070663824146360538192, 3.83208359418711170321037230257, 4.82663288990400200190820024985, 4.92092142676527129987847930432, 5.46363929357380701737020358050, 5.58427485792964587314152582088, 5.97728761123171735345903077564, 6.02494145781311797519071895620, 6.80882354276778293913162940303, 7.30671405304916208092400729211, 7.46473132756715875357590270316, 7.66775686795789806282976278921
