L(s) = 1 | − 1.73·3-s + 5-s + 1.99·9-s + 1.73·11-s − 13-s − 1.73·15-s − 1.73·27-s + 29-s − 1.73·31-s − 2.99·33-s + 1.73·39-s + 1.73·43-s + 1.99·45-s + 1.73·47-s + 49-s − 53-s + 1.73·55-s − 65-s + 1.73·79-s + 0.999·81-s − 1.73·87-s + 2.99·93-s + 3.46·99-s + 109-s − 1.99·117-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 5-s + 1.99·9-s + 1.73·11-s − 13-s − 1.73·15-s − 1.73·27-s + 29-s − 1.73·31-s − 2.99·33-s + 1.73·39-s + 1.73·43-s + 1.99·45-s + 1.73·47-s + 49-s − 53-s + 1.73·55-s − 65-s + 1.73·79-s + 0.999·81-s − 1.73·87-s + 2.99·93-s + 3.46·99-s + 109-s − 1.99·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8332460942\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8332460942\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + 1.73T + T^{2} \) |
| 5 | \( 1 - T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.73T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 - 1.73T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.480582269182632844074852204446, −9.060908692437951770353606492523, −7.47164182246827528528747375725, −6.82850415698046821562939409831, −6.09566633705256684734347064283, −5.61225228812785390847597822983, −4.72413157490766022001646151037, −3.89991009108918121733817148876, −2.17239736891688784877413198971, −1.06655896931933305489203305438,
1.06655896931933305489203305438, 2.17239736891688784877413198971, 3.89991009108918121733817148876, 4.72413157490766022001646151037, 5.61225228812785390847597822983, 6.09566633705256684734347064283, 6.82850415698046821562939409831, 7.47164182246827528528747375725, 9.060908692437951770353606492523, 9.480582269182632844074852204446