| L(s) = 1 | + 2.80·2-s + 0.342·3-s + 5.84·4-s + 0.958·6-s − 1.04·7-s + 10.7·8-s − 2.88·9-s + 4.64·11-s + 1.99·12-s − 2.95·13-s − 2.91·14-s + 18.4·16-s − 6.29·17-s − 8.07·18-s − 1.11·19-s − 0.356·21-s + 12.9·22-s − 3.87·23-s + 3.68·24-s − 8.28·26-s − 2.01·27-s − 6.08·28-s + 2.35·29-s + 31-s + 30.1·32-s + 1.58·33-s − 17.6·34-s + ⋯ |
| L(s) = 1 | + 1.98·2-s + 0.197·3-s + 2.92·4-s + 0.391·6-s − 0.393·7-s + 3.80·8-s − 0.960·9-s + 1.39·11-s + 0.577·12-s − 0.820·13-s − 0.779·14-s + 4.61·16-s − 1.52·17-s − 1.90·18-s − 0.255·19-s − 0.0777·21-s + 2.77·22-s − 0.807·23-s + 0.751·24-s − 1.62·26-s − 0.387·27-s − 1.14·28-s + 0.437·29-s + 0.179·31-s + 5.32·32-s + 0.276·33-s − 3.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.143921072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.143921072\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 3 | \( 1 - 0.342T + 3T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 + 6.29T + 17T^{2} \) |
| 19 | \( 1 + 1.11T + 19T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 37 | \( 1 + 1.30T + 37T^{2} \) |
| 41 | \( 1 - 1.92T + 41T^{2} \) |
| 43 | \( 1 - 4.29T + 43T^{2} \) |
| 47 | \( 1 - 3.31T + 47T^{2} \) |
| 53 | \( 1 + 5.10T + 53T^{2} \) |
| 59 | \( 1 + 5.07T + 59T^{2} \) |
| 61 | \( 1 - 5.31T + 61T^{2} \) |
| 67 | \( 1 - 8.64T + 67T^{2} \) |
| 71 | \( 1 + 3.88T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 - 7.46T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 0.915T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76602903158039062410901463007, −9.602174260391766021319847249023, −8.442470563033910035342043351096, −7.23057097913840350892763533725, −6.46594836957735520020464806887, −5.89208456569144413429330069267, −4.70105145711983626468779465300, −3.99700187440753013310003056199, −2.95870501423024019484989959932, −2.03040231961256133857767186799,
2.03040231961256133857767186799, 2.95870501423024019484989959932, 3.99700187440753013310003056199, 4.70105145711983626468779465300, 5.89208456569144413429330069267, 6.46594836957735520020464806887, 7.23057097913840350892763533725, 8.442470563033910035342043351096, 9.602174260391766021319847249023, 10.76602903158039062410901463007
