| L(s) = 1 | + 5.05e5·3-s + 9.01e7·5-s + 6.87e9·7-s + 1.61e11·9-s + 9.65e11·11-s − 5.42e11·13-s + 4.56e13·15-s + 8.20e13·17-s − 5.55e14·19-s + 3.47e15·21-s + 6.50e15·23-s − 3.79e15·25-s + 3.42e16·27-s + 1.22e16·29-s + 1.19e17·31-s + 4.88e17·33-s + 6.19e17·35-s + 6.19e17·37-s − 2.74e17·39-s − 1.58e18·41-s − 8.37e18·43-s + 1.45e19·45-s + 1.31e19·47-s + 1.98e19·49-s + 4.15e19·51-s − 4.17e19·53-s + 8.70e19·55-s + ⋯ |
| L(s) = 1 | + 1.64·3-s + 0.825·5-s + 1.31·7-s + 1.71·9-s + 1.02·11-s − 0.0839·13-s + 1.36·15-s + 0.580·17-s − 1.09·19-s + 2.16·21-s + 1.42·23-s − 0.318·25-s + 1.18·27-s + 0.185·29-s + 0.848·31-s + 1.68·33-s + 1.08·35-s + 0.572·37-s − 0.138·39-s − 0.450·41-s − 1.37·43-s + 1.41·45-s + 0.772·47-s + 0.725·49-s + 0.957·51-s − 0.619·53-s + 0.842·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(7.658670538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.658670538\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 56212 p^{2} T + p^{23} T^{2} \) |
| 5 | \( 1 - 18027114 p T + p^{23} T^{2} \) |
| 7 | \( 1 - 140250104 p^{2} T + p^{23} T^{2} \) |
| 11 | \( 1 - 87757163508 p T + p^{23} T^{2} \) |
| 13 | \( 1 + 41719999934 p T + p^{23} T^{2} \) |
| 17 | \( 1 - 82083537265266 T + p^{23} T^{2} \) |
| 19 | \( 1 + 29249923769300 p T + p^{23} T^{2} \) |
| 23 | \( 1 - 6508638190765032 T + p^{23} T^{2} \) |
| 29 | \( 1 - 12202037915600490 T + p^{23} T^{2} \) |
| 31 | \( 1 - 119978011042749152 T + p^{23} T^{2} \) |
| 37 | \( 1 - 619510980267421234 T + p^{23} T^{2} \) |
| 41 | \( 1 + 1587735553771936038 T + p^{23} T^{2} \) |
| 43 | \( 1 + 8377717142038508132 T + p^{23} T^{2} \) |
| 47 | \( 1 - 13100457020745462096 T + p^{23} T^{2} \) |
| 53 | \( 1 + 41795979279875033022 T + p^{23} T^{2} \) |
| 59 | \( 1 - 74383865281405054380 T + p^{23} T^{2} \) |
| 61 | \( 1 - \)\(27\!\cdots\!98\)\( T + p^{23} T^{2} \) |
| 67 | \( 1 + \)\(17\!\cdots\!76\)\( T + p^{23} T^{2} \) |
| 71 | \( 1 + \)\(27\!\cdots\!68\)\( T + p^{23} T^{2} \) |
| 73 | \( 1 - \)\(43\!\cdots\!62\)\( T + p^{23} T^{2} \) |
| 79 | \( 1 - \)\(35\!\cdots\!40\)\( T + p^{23} T^{2} \) |
| 83 | \( 1 - \)\(22\!\cdots\!28\)\( T + p^{23} T^{2} \) |
| 89 | \( 1 - \)\(33\!\cdots\!10\)\( T + p^{23} T^{2} \) |
| 97 | \( 1 - \)\(92\!\cdots\!06\)\( T + p^{23} T^{2} \) |
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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
−10.38725787706524125405326571239, −9.317885587791614091829198007108, −8.593135200569066678532519388770, −7.74062248976841454132722316955, −6.50780855970694459170254407939, −4.96897639293333171665759348949, −3.93685397665069309819016216367, −2.75139457410941996081678262001, −1.83202975094128981741245118976, −1.20297020203097517696119903862,
1.20297020203097517696119903862, 1.83202975094128981741245118976, 2.75139457410941996081678262001, 3.93685397665069309819016216367, 4.96897639293333171665759348949, 6.50780855970694459170254407939, 7.74062248976841454132722316955, 8.593135200569066678532519388770, 9.317885587791614091829198007108, 10.38725787706524125405326571239
