| L(s) = 1 | + 2-s − 3-s + 4-s + 3.49·5-s − 6-s − 1.05·7-s + 8-s + 9-s + 3.49·10-s + 4.23·11-s − 12-s + 5.43·13-s − 1.05·14-s − 3.49·15-s + 16-s − 3.65·17-s + 18-s − 1.05·19-s + 3.49·20-s + 1.05·21-s + 4.23·22-s + 23-s − 24-s + 7.21·25-s + 5.43·26-s − 27-s − 1.05·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.56·5-s − 0.408·6-s − 0.399·7-s + 0.353·8-s + 0.333·9-s + 1.10·10-s + 1.27·11-s − 0.288·12-s + 1.50·13-s − 0.282·14-s − 0.902·15-s + 0.250·16-s − 0.887·17-s + 0.235·18-s − 0.242·19-s + 0.781·20-s + 0.230·21-s + 0.903·22-s + 0.208·23-s − 0.204·24-s + 1.44·25-s + 1.06·26-s − 0.192·27-s − 0.199·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.852021326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.852021326\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 3.49T + 5T^{2} \) |
| 7 | \( 1 + 1.05T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 - 5.43T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 1.05T + 19T^{2} \) |
| 31 | \( 1 + 4.03T + 31T^{2} \) |
| 37 | \( 1 - 7.11T + 37T^{2} \) |
| 41 | \( 1 - 4.28T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 4.96T + 47T^{2} \) |
| 53 | \( 1 - 9.06T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 9.18T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 4.66T + 79T^{2} \) |
| 83 | \( 1 + 4.60T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 4.83T + 97T^{2} \) |
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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
−8.718957115725790251562882155576, −7.38221378881649191888927957758, −6.50279410499102566864086594319, −6.11815993859787658002547032110, −5.81636428238400712831404758654, −4.67415891929410516698011682525, −4.01144486594651197531888680076, −2.99835264138898912000984326483, −1.90225615557712198814826565915, −1.16256757955017448808660945585,
1.16256757955017448808660945585, 1.90225615557712198814826565915, 2.99835264138898912000984326483, 4.01144486594651197531888680076, 4.67415891929410516698011682525, 5.81636428238400712831404758654, 6.11815993859787658002547032110, 6.50279410499102566864086594319, 7.38221378881649191888927957758, 8.718957115725790251562882155576
