| L(s) = 1 | + 3-s − 2·5-s − 7-s − 9-s − 2·11-s − 4·13-s − 2·15-s + 3·17-s − 19-s − 21-s + 2·23-s + 3·25-s − 7·29-s + 9·31-s − 2·33-s + 2·35-s − 37-s − 4·39-s + 4·41-s − 4·43-s + 2·45-s − 18·47-s − 9·49-s + 3·51-s − 5·53-s + 4·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s − 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.516·15-s + 0.727·17-s − 0.229·19-s − 0.218·21-s + 0.417·23-s + 3/5·25-s − 1.29·29-s + 1.61·31-s − 0.348·33-s + 0.338·35-s − 0.164·37-s − 0.640·39-s + 0.624·41-s − 0.609·43-s + 0.298·45-s − 2.62·47-s − 9/7·49-s + 0.420·51-s − 0.686·53-s + 0.539·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 32 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 108 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 17 T + 156 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 150 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 284 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 186 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.263043210712759395669160396547, −8.062638983705104623543178968024, −7.58411333143635315277547676064, −7.55073798164263527200502167559, −6.94014429261237884810882615735, −6.65214207653145472809896282566, −6.13502205718457933088739792573, −5.85249471551881949555722066619, −5.15521075165665950837055366635, −4.94648615064149947829265339892, −4.60446901847424552865119536698, −4.12324400852933763853019697987, −3.51465603824553589679406975904, −3.20341576136857116286814121413, −2.83948669895377980384520592448, −2.54647004983574340309073146997, −1.74037358281045983745569225727, −1.22277422682839801850135116434, 0, 0,
1.22277422682839801850135116434, 1.74037358281045983745569225727, 2.54647004983574340309073146997, 2.83948669895377980384520592448, 3.20341576136857116286814121413, 3.51465603824553589679406975904, 4.12324400852933763853019697987, 4.60446901847424552865119536698, 4.94648615064149947829265339892, 5.15521075165665950837055366635, 5.85249471551881949555722066619, 6.13502205718457933088739792573, 6.65214207653145472809896282566, 6.94014429261237884810882615735, 7.55073798164263527200502167559, 7.58411333143635315277547676064, 8.062638983705104623543178968024, 8.263043210712759395669160396547
