| L(s) = 1 | − 8·2-s + 48·4-s + 14·5-s − 256·8-s − 112·10-s + 238·13-s + 1.28e3·16-s − 322·17-s + 672·20-s + 625·25-s − 1.90e3·26-s − 82·29-s − 6.14e3·32-s + 2.57e3·34-s − 2.16e3·37-s − 3.58e3·40-s − 6.07e3·41-s + 4.80e3·49-s − 5.00e3·50-s + 1.14e4·52-s + 4.96e3·53-s + 656·58-s + 6.95e3·61-s + 2.86e4·64-s + 3.33e3·65-s − 1.54e4·68-s − 1.44e3·73-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s + 0.559·5-s − 4·8-s − 1.11·10-s + 1.40·13-s + 5·16-s − 1.11·17-s + 1.67·20-s + 25-s − 2.81·26-s − 0.0975·29-s − 6·32-s + 2.22·34-s − 1.57·37-s − 2.23·40-s − 3.61·41-s + 2·49-s − 2·50-s + 4.22·52-s + 1.76·53-s + 0.195·58-s + 1.86·61-s + 7·64-s + 0.788·65-s − 3.34·68-s − 0.270·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.095643182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.095643182\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 14 T - 429 T^{2} - 14 p^{4} T^{3} + p^{8} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 238 T + 28083 T^{2} - 238 p^{4} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 322 T + 20163 T^{2} + 322 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 82 T - 700557 T^{2} + 82 p^{4} T^{3} + p^{8} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2162 T + 2800083 T^{2} + 2162 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3038 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2482 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 6958 T + 34567923 T^{2} - 6958 p^{4} T^{3} + p^{8} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1442 T - 26318877 T^{2} + 1442 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 9758 T + 32476323 T^{2} - 9758 p^{4} T^{3} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1918 T + p^{4} T^{2} )^{2} \) |
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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
−10.92098453619654129324993382638, −10.62334214475040727697455764627, −10.12442679026966955024074756678, −10.02188274594968867943812499691, −9.133383859167122519586882748123, −8.788607130634923334771966391226, −8.492793515532948003022060905934, −8.361722259383735660373348009406, −7.20197629839181424349973872241, −7.09501987506146510675870471692, −6.62754972470348836073928023126, −6.04329805288691216851480862426, −5.58628726398014380089382156595, −4.88125148192456752829498939112, −3.60856115146991844850725248526, −3.35178008298040332936250387489, −2.24643002691314279694663515114, −1.96293514357086835431005206468, −1.14038322005383908360935438989, −0.48216546923391136954581502350,
0.48216546923391136954581502350, 1.14038322005383908360935438989, 1.96293514357086835431005206468, 2.24643002691314279694663515114, 3.35178008298040332936250387489, 3.60856115146991844850725248526, 4.88125148192456752829498939112, 5.58628726398014380089382156595, 6.04329805288691216851480862426, 6.62754972470348836073928023126, 7.09501987506146510675870471692, 7.20197629839181424349973872241, 8.361722259383735660373348009406, 8.492793515532948003022060905934, 8.788607130634923334771966391226, 9.133383859167122519586882748123, 10.02188274594968867943812499691, 10.12442679026966955024074756678, 10.62334214475040727697455764627, 10.92098453619654129324993382638
