| 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
| \(L(2)\) |
\(\approx\) |
\(1.931714027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.931714027\) |
| \(L(\frac{5}{2})\) |
|
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
−14.23534341828736522045330952780, −13.20371472894862286672934778707, −12.76744530243510483156187460648, −12.14999325219835497599682072968, −11.49620874362813336611085072604, −11.12883665890889697133001382975, −10.79692042750257337933759432529, −9.847055857361340957179909248261, −9.703584351821489114195542910520, −8.621210715449709086793515304330, −8.483499966845826358509697583367, −6.88775802726927260919374450678, −6.66969774693845423182921770203, −5.95097404480670622678660658443, −5.81582514513634347794317059811, −4.63025257367326125683904459209, −4.62893088822233280156749678283, −3.26867368760505626465876652682, −2.04158525727052133325109303156, −0.818734667486362094071737159048,
0.818734667486362094071737159048, 2.04158525727052133325109303156, 3.26867368760505626465876652682, 4.62893088822233280156749678283, 4.63025257367326125683904459209, 5.81582514513634347794317059811, 5.95097404480670622678660658443, 6.66969774693845423182921770203, 6.88775802726927260919374450678, 8.483499966845826358509697583367, 8.621210715449709086793515304330, 9.703584351821489114195542910520, 9.847055857361340957179909248261, 10.79692042750257337933759432529, 11.12883665890889697133001382975, 11.49620874362813336611085072604, 12.14999325219835497599682072968, 12.76744530243510483156187460648, 13.20371472894862286672934778707, 14.23534341828736522045330952780
