| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s + 16-s + 17-s − 19-s − 22-s − 32-s − 34-s + 38-s + 41-s + 2·43-s + 44-s + 49-s − 2·59-s + 64-s − 67-s + 68-s − 73-s − 76-s − 82-s + 83-s − 2·86-s − 88-s + 89-s + 2·97-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s + 16-s + 17-s − 19-s − 22-s − 32-s − 34-s + 38-s + 41-s + 2·43-s + 44-s + 49-s − 2·59-s + 64-s − 67-s + 68-s − 73-s − 76-s − 82-s + 83-s − 2·86-s − 88-s + 89-s + 2·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7935611990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7935611990\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 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
−9.182752917230046270758357640667, −8.973760283098123617812376239629, −7.85430724067572317511676325756, −7.34552662428788505358233760198, −6.31024208825912204894174527701, −5.83037375984899647919832171477, −4.42419972209165543874850193681, −3.40739747936208947909986320004, −2.28349598699875889294751975140, −1.11023952841489479655423604656,
1.11023952841489479655423604656, 2.28349598699875889294751975140, 3.40739747936208947909986320004, 4.42419972209165543874850193681, 5.83037375984899647919832171477, 6.31024208825912204894174527701, 7.34552662428788505358233760198, 7.85430724067572317511676325756, 8.973760283098123617812376239629, 9.182752917230046270758357640667
