| L(s) = 1 | + 1.14e5·5-s + 3.03e6·7-s − 1.03e8·11-s − 1.04e8·13-s − 9.97e8·17-s − 4.93e9·19-s + 8.32e9·23-s − 1.73e10·25-s − 1.04e11·29-s + 2.96e11·31-s + 3.48e11·35-s − 1.78e11·37-s + 1.79e12·41-s + 2.86e12·43-s + 4.33e12·47-s + 4.46e12·49-s − 9.73e12·53-s − 1.18e13·55-s − 1.35e13·59-s + 5.35e12·61-s − 1.19e13·65-s + 5.32e13·67-s − 2.02e13·71-s + 2.62e13·73-s − 3.13e14·77-s + 3.39e14·79-s + 1.31e14·83-s + ⋯ |
| L(s) = 1 | + 0.657·5-s + 1.39·7-s − 1.60·11-s − 0.461·13-s − 0.589·17-s − 1.26·19-s + 0.509·23-s − 0.568·25-s − 1.12·29-s + 1.93·31-s + 0.915·35-s − 0.308·37-s + 1.43·41-s + 1.60·43-s + 1.24·47-s + 0.939·49-s − 1.13·53-s − 1.05·55-s − 0.707·59-s + 0.218·61-s − 0.303·65-s + 1.07·67-s − 0.263·71-s + 0.278·73-s − 2.22·77-s + 1.98·79-s + 0.532·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(2.472378977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.472378977\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 22962 p T + p^{15} T^{2} \) |
| 7 | \( 1 - 433504 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 9404700 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 104365834 T + p^{15} T^{2} \) |
| 17 | \( 1 + 997689762 T + p^{15} T^{2} \) |
| 19 | \( 1 + 4934015444 T + p^{15} T^{2} \) |
| 23 | \( 1 - 8324920200 T + p^{15} T^{2} \) |
| 29 | \( 1 + 104128242846 T + p^{15} T^{2} \) |
| 31 | \( 1 - 296696681512 T + p^{15} T^{2} \) |
| 37 | \( 1 + 178337455666 T + p^{15} T^{2} \) |
| 41 | \( 1 - 1790882416086 T + p^{15} T^{2} \) |
| 43 | \( 1 - 2863459422772 T + p^{15} T^{2} \) |
| 47 | \( 1 - 4332907521600 T + p^{15} T^{2} \) |
| 53 | \( 1 + 9732317104422 T + p^{15} T^{2} \) |
| 59 | \( 1 + 13514837176500 T + p^{15} T^{2} \) |
| 61 | \( 1 - 5352663511190 T + p^{15} T^{2} \) |
| 67 | \( 1 - 53233909720108 T + p^{15} T^{2} \) |
| 71 | \( 1 + 20229661643400 T + p^{15} T^{2} \) |
| 73 | \( 1 - 26264166466106 T + p^{15} T^{2} \) |
| 79 | \( 1 - 339031361615128 T + p^{15} T^{2} \) |
| 83 | \( 1 - 131684771045076 T + p^{15} T^{2} \) |
| 89 | \( 1 - 39352148322678 T + p^{15} T^{2} \) |
| 97 | \( 1 - 1128750908801474 T + p^{15} T^{2} \) |
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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
−10.58169957304831604321852006016, −9.361522731696875851750093441092, −8.217301284436579171664811123390, −7.53779673080235708207915405936, −6.09121378514327329150639770725, −5.11811366457347034772587064304, −4.34739589993904123754056114751, −2.51928256003195110724631698406, −2.01479717269964827477934748136, −0.63648785934153574203685333581,
0.63648785934153574203685333581, 2.01479717269964827477934748136, 2.51928256003195110724631698406, 4.34739589993904123754056114751, 5.11811366457347034772587064304, 6.09121378514327329150639770725, 7.53779673080235708207915405936, 8.217301284436579171664811123390, 9.361522731696875851750093441092, 10.58169957304831604321852006016
