| L(s) = 1 | − 4·4-s − 9·9-s − 138·11-s + 16·16-s − 112·19-s + 66·29-s − 140·31-s + 36·36-s + 1.00e3·41-s + 552·44-s − 49·49-s − 96·59-s + 1.04e3·61-s − 64·64-s − 186·71-s + 448·76-s + 2.52e3·79-s + 81·81-s − 600·89-s + 1.24e3·99-s + 1.33e3·101-s − 1.16e3·109-s − 264·116-s + 1.16e4·121-s + 560·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 3.78·11-s + 1/4·16-s − 1.35·19-s + 0.422·29-s − 0.811·31-s + 1/6·36-s + 3.83·41-s + 1.89·44-s − 1/7·49-s − 0.211·59-s + 2.19·61-s − 1/8·64-s − 0.310·71-s + 0.676·76-s + 3.59·79-s + 1/9·81-s − 0.714·89-s + 1.26·99-s + 1.31·101-s − 1.02·109-s − 0.211·116-s + 8.73·121-s + 0.405·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.133620197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.133620197\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 69 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 298 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3170 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 56 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 24253 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 33 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 98497 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 504 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 140245 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 152354 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 27146 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 524 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 601525 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 93 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 55534 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1261 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 907378 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 300 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1075390 T^{2} + p^{6} T^{4} \) |
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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
−9.661637950747938443537104145042, −9.444729847211675234307470951089, −8.857341933145233369821890195540, −8.343340051597186025180610342130, −8.005438331475640490894601952422, −7.903667955268404482761534714857, −7.37324063459405926417347400131, −6.93321043912215866406301452886, −6.18383945675224575277434568784, −5.64367334326986633938233843893, −5.60711143869937824432250935070, −4.93318238433739626154743867544, −4.66245464643627936920265129551, −4.09641472364550540800017828037, −3.40761293132974369897198851544, −2.80030799109411035490984610998, −2.27805272865437588480317121935, −2.21060127833579740312443087626, −0.71826263048227495086692345113, −0.38137538291714239930680357081,
0.38137538291714239930680357081, 0.71826263048227495086692345113, 2.21060127833579740312443087626, 2.27805272865437588480317121935, 2.80030799109411035490984610998, 3.40761293132974369897198851544, 4.09641472364550540800017828037, 4.66245464643627936920265129551, 4.93318238433739626154743867544, 5.60711143869937824432250935070, 5.64367334326986633938233843893, 6.18383945675224575277434568784, 6.93321043912215866406301452886, 7.37324063459405926417347400131, 7.903667955268404482761534714857, 8.005438331475640490894601952422, 8.343340051597186025180610342130, 8.857341933145233369821890195540, 9.444729847211675234307470951089, 9.661637950747938443537104145042
