| L(s) = 1 | + 2·7-s − 4·19-s + 4·25-s + 12·29-s − 16·31-s − 8·37-s − 24·47-s − 3·49-s − 12·53-s − 24·59-s + 32·103-s − 4·109-s − 12·113-s + 16·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 8·175-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.917·19-s + 4/5·25-s + 2.22·29-s − 2.87·31-s − 1.31·37-s − 3.50·47-s − 3/7·49-s − 1.64·53-s − 3.12·59-s + 3.15·103-s − 0.383·109-s − 1.12·113-s + 1.45·121-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + 0.604·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9964831613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9964831613\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 170 T^{2} + p^{2} T^{4} \) |
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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
−8.708593713111611465755588759321, −8.232924760932119366334810666706, −7.86256143244287086622108061194, −7.64945118094453465598096252766, −7.23593100646273475237048327011, −6.62846549715368849129276923782, −6.36346463538286428002508950710, −6.35218317129950919295603072176, −5.57869474226415626388316960054, −5.03479101598145022740793529849, −4.98660763950929403122416588415, −4.56707700992666004285186064587, −4.18896662749837458752363235920, −3.36375950647403151529675775088, −3.32900047416506347700725316131, −2.84277589082081442105851650956, −2.01755300674500948562149542824, −1.67874550249301795228447373847, −1.38814178579189671761878256703, −0.27610689699380495329889963903,
0.27610689699380495329889963903, 1.38814178579189671761878256703, 1.67874550249301795228447373847, 2.01755300674500948562149542824, 2.84277589082081442105851650956, 3.32900047416506347700725316131, 3.36375950647403151529675775088, 4.18896662749837458752363235920, 4.56707700992666004285186064587, 4.98660763950929403122416588415, 5.03479101598145022740793529849, 5.57869474226415626388316960054, 6.35218317129950919295603072176, 6.36346463538286428002508950710, 6.62846549715368849129276923782, 7.23593100646273475237048327011, 7.64945118094453465598096252766, 7.86256143244287086622108061194, 8.232924760932119366334810666706, 8.708593713111611465755588759321
