L(s) = 1 | + 1.05·2-s − 0.687·3-s − 0.881·4-s − 0.726·6-s + 1.01·7-s − 3.04·8-s − 2.52·9-s + 5.12·11-s + 0.606·12-s + 6.08·13-s + 1.07·14-s − 1.45·16-s + 3.19·17-s − 2.67·18-s + 3.42·19-s − 0.695·21-s + 5.41·22-s + 2.91·23-s + 2.09·24-s + 6.43·26-s + 3.79·27-s − 0.892·28-s − 1.55·29-s − 7.99·31-s + 4.55·32-s − 3.52·33-s + 3.37·34-s + ⋯ |
L(s) = 1 | + 0.747·2-s − 0.396·3-s − 0.440·4-s − 0.296·6-s + 0.382·7-s − 1.07·8-s − 0.842·9-s + 1.54·11-s + 0.175·12-s + 1.68·13-s + 0.285·14-s − 0.364·16-s + 0.774·17-s − 0.629·18-s + 0.786·19-s − 0.151·21-s + 1.15·22-s + 0.608·23-s + 0.427·24-s + 1.26·26-s + 0.731·27-s − 0.168·28-s − 0.288·29-s − 1.43·31-s + 0.804·32-s − 0.612·33-s + 0.579·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.763074154\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.763074154\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 - 1.05T + 2T^{2} \) |
| 3 | \( 1 + 0.687T + 3T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 - 6.08T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 - 3.42T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 + 7.99T + 31T^{2} \) |
| 37 | \( 1 - 8.40T + 37T^{2} \) |
| 41 | \( 1 + 1.86T + 41T^{2} \) |
| 43 | \( 1 + 5.22T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 2.89T + 59T^{2} \) |
| 61 | \( 1 + 2.30T + 61T^{2} \) |
| 67 | \( 1 - 4.64T + 67T^{2} \) |
| 71 | \( 1 + 7.73T + 71T^{2} \) |
| 73 | \( 1 + 0.595T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 6.39T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−10.99931087035606606246820565791, −9.559393528384947677719906855012, −8.914387226864149752226089431729, −8.130916851684908282543829358301, −6.66554862766341716596777195181, −5.87511061439096925094517011522, −5.19351292340122704813651872950, −3.94729036029579472021414778743, −3.28616281469534592023672412973, −1.17101973537407762569401841610,
1.17101973537407762569401841610, 3.28616281469534592023672412973, 3.94729036029579472021414778743, 5.19351292340122704813651872950, 5.87511061439096925094517011522, 6.66554862766341716596777195181, 8.130916851684908282543829358301, 8.914387226864149752226089431729, 9.559393528384947677719906855012, 10.99931087035606606246820565791