| L(s) = 1 | + 2·2-s + 8·3-s − 14·5-s + 16·6-s − 8·8-s + 27·9-s − 28·10-s + 28·11-s − 36·13-s − 112·15-s − 16·16-s + 74·17-s + 54·18-s + 80·19-s + 56·22-s + 112·23-s − 64·24-s + 125·25-s − 72·26-s + 136·27-s + 380·29-s − 224·30-s + 72·31-s + 224·33-s + 148·34-s + 346·37-s + 160·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s − 1.25·5-s + 1.08·6-s − 0.353·8-s + 9-s − 0.885·10-s + 0.767·11-s − 0.768·13-s − 1.92·15-s − 1/4·16-s + 1.05·17-s + 0.707·18-s + 0.965·19-s + 0.542·22-s + 1.01·23-s − 0.544·24-s + 25-s − 0.543·26-s + 0.969·27-s + 2.43·29-s − 1.36·30-s + 0.417·31-s + 1.18·33-s + 0.746·34-s + 1.53·37-s + 0.683·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.932666028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.932666028\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 8 T + 37 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 14 T + 71 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 28 T - 547 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 18 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 74 T + 563 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 80 T - 459 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 112 T + 377 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 190 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 72 T - 24607 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 346 T + 69063 T^{2} - 346 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 162 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 412 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 24 T - 103247 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 p T - 17 p^{2} T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 200 T - 165379 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 198 T - 187777 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 716 T + 211893 T^{2} - 716 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 392 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 538 T - 99573 T^{2} - 538 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 240 T - 435439 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 1072 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 810 T - 48869 T^{2} - 810 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1354 T + p^{3} T^{2} )^{2} \) |
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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
−14.00614845287849239959170536340, −13.43825133410521733206931973599, −12.48413438363811637156813089276, −12.28619358027130845703390165533, −11.82193364760982477356258311837, −11.27416302320278753213136565612, −10.40619423012854610852413114251, −9.659363550150852267906323514965, −9.427703125486913488613248172341, −8.402363262280175094023732650677, −8.289531823673146122024074884486, −7.76833610046619571083999598134, −6.85311550859679109674657952767, −6.53642740953449782102107490720, −4.98718529711639976822465318950, −4.83372578159566528961249495196, −3.74185215959515808712540253060, −3.23445181065166001337830034573, −2.74235385625171322885891348375, −1.03833618770149882262981493728,
1.03833618770149882262981493728, 2.74235385625171322885891348375, 3.23445181065166001337830034573, 3.74185215959515808712540253060, 4.83372578159566528961249495196, 4.98718529711639976822465318950, 6.53642740953449782102107490720, 6.85311550859679109674657952767, 7.76833610046619571083999598134, 8.289531823673146122024074884486, 8.402363262280175094023732650677, 9.427703125486913488613248172341, 9.659363550150852267906323514965, 10.40619423012854610852413114251, 11.27416302320278753213136565612, 11.82193364760982477356258311837, 12.28619358027130845703390165533, 12.48413438363811637156813089276, 13.43825133410521733206931973599, 14.00614845287849239959170536340
