| L(s) = 1 | + 4·5-s − 7-s + 6·11-s − 6·17-s + 8·19-s − 8·23-s + 11·25-s − 2·29-s + 4·31-s − 4·35-s + 2·37-s − 4·41-s − 4·43-s − 10·47-s + 49-s − 6·53-s + 24·55-s + 6·59-s + 2·61-s + 12·67-s − 6·71-s + 10·73-s − 6·77-s − 8·79-s − 14·83-s − 24·85-s + 32·95-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s + 1.80·11-s − 1.45·17-s + 1.83·19-s − 1.66·23-s + 11/5·25-s − 0.371·29-s + 0.718·31-s − 0.676·35-s + 0.328·37-s − 0.624·41-s − 0.609·43-s − 1.45·47-s + 1/7·49-s − 0.824·53-s + 3.23·55-s + 0.781·59-s + 0.256·61-s + 1.46·67-s − 0.712·71-s + 1.17·73-s − 0.683·77-s − 0.900·79-s − 1.53·83-s − 2.60·85-s + 3.28·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.147801026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.147801026\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.84102109796311, −13.58969521010483, −13.06750248366096, −12.54240237013885, −11.84859191065484, −11.50826935447561, −11.05138976361687, −10.07479225592817, −9.878862808214139, −9.570750086204922, −9.046412736850590, −8.574720986337914, −7.923390438064101, −6.977259997500003, −6.649532782310921, −6.319003723147527, −5.739912754247841, −5.237854616614588, −4.539440210704753, −3.925988915000284, −3.251181169143492, −2.601094188421291, −1.756504226508544, −1.596409703560530, −0.6444057955677406,
0.6444057955677406, 1.596409703560530, 1.756504226508544, 2.601094188421291, 3.251181169143492, 3.925988915000284, 4.539440210704753, 5.237854616614588, 5.739912754247841, 6.319003723147527, 6.649532782310921, 6.977259997500003, 7.923390438064101, 8.574720986337914, 9.046412736850590, 9.570750086204922, 9.878862808214139, 10.07479225592817, 11.05138976361687, 11.50826935447561, 11.84859191065484, 12.54240237013885, 13.06750248366096, 13.58969521010483, 13.84102109796311
