| L(s) = 1 | + 5·5-s + 8·7-s + 4·11-s − 6·13-s + 2·17-s + 16·19-s − 60·23-s + 25·25-s + 142·29-s + 176·31-s + 40·35-s − 214·37-s + 278·41-s + 68·43-s + 116·47-s − 279·49-s + 350·53-s + 20·55-s + 684·59-s − 394·61-s − 30·65-s − 108·67-s − 96·71-s − 398·73-s + 32·77-s − 136·79-s + 436·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.431·7-s + 0.109·11-s − 0.128·13-s + 0.0285·17-s + 0.193·19-s − 0.543·23-s + 1/5·25-s + 0.909·29-s + 1.01·31-s + 0.193·35-s − 0.950·37-s + 1.05·41-s + 0.241·43-s + 0.360·47-s − 0.813·49-s + 0.907·53-s + 0.0490·55-s + 1.50·59-s − 0.826·61-s − 0.0572·65-s − 0.196·67-s − 0.160·71-s − 0.638·73-s + 0.0473·77-s − 0.193·79-s + 0.576·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.551123183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.551123183\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 - 16 T + p^{3} T^{2} \) |
| 23 | \( 1 + 60 T + p^{3} T^{2} \) |
| 29 | \( 1 - 142 T + p^{3} T^{2} \) |
| 31 | \( 1 - 176 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 278 T + p^{3} T^{2} \) |
| 43 | \( 1 - 68 T + p^{3} T^{2} \) |
| 47 | \( 1 - 116 T + p^{3} T^{2} \) |
| 53 | \( 1 - 350 T + p^{3} T^{2} \) |
| 59 | \( 1 - 684 T + p^{3} T^{2} \) |
| 61 | \( 1 + 394 T + p^{3} T^{2} \) |
| 67 | \( 1 + 108 T + p^{3} T^{2} \) |
| 71 | \( 1 + 96 T + p^{3} T^{2} \) |
| 73 | \( 1 + 398 T + p^{3} T^{2} \) |
| 79 | \( 1 + 136 T + p^{3} T^{2} \) |
| 83 | \( 1 - 436 T + p^{3} T^{2} \) |
| 89 | \( 1 - 750 T + p^{3} T^{2} \) |
| 97 | \( 1 - 82 T + p^{3} T^{2} \) |
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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
−9.131768526814424120851624440208, −8.396605074421235903446078454751, −7.59221297714023975808421267711, −6.67765118949623631963326555762, −5.87391649775497196484658335616, −4.99185961046917821845735536783, −4.14289445596208043436685320015, −2.94638384658623576411236088630, −1.94546405213263710566367196803, −0.804821176280024084810123713662,
0.804821176280024084810123713662, 1.94546405213263710566367196803, 2.94638384658623576411236088630, 4.14289445596208043436685320015, 4.99185961046917821845735536783, 5.87391649775497196484658335616, 6.67765118949623631963326555762, 7.59221297714023975808421267711, 8.396605074421235903446078454751, 9.131768526814424120851624440208
