| L(s) = 1 | + 3-s + 5-s + 9-s + 11-s + 6·13-s + 15-s + 17-s − 4·19-s + 25-s + 27-s + 6·29-s + 33-s − 2·37-s + 6·39-s − 6·41-s + 4·43-s + 45-s + 8·47-s − 7·49-s + 51-s + 14·53-s + 55-s − 4·57-s − 12·59-s − 2·61-s + 6·65-s − 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.258·15-s + 0.242·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.174·33-s − 0.328·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s − 49-s + 0.140·51-s + 1.92·53-s + 0.134·55-s − 0.529·57-s − 1.56·59-s − 0.256·61-s + 0.744·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.130569578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.130569578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.50546018511346, −14.12745895252577, −13.54489481631051, −13.34969204975736, −12.63366846960862, −12.15715069836205, −11.54732693065998, −10.86066842768618, −10.46530650687727, −10.02774828589883, −9.225262316772368, −8.798999097815738, −8.468746383456718, −7.867724491235969, −7.116979017774687, −6.550535996952960, −6.063364069977190, −5.555358639378547, −4.632217587565054, −4.162159063883735, −3.462428634579362, −2.941076318462707, −2.086176572214125, −1.483542131179036, −0.7323719205649915,
0.7323719205649915, 1.483542131179036, 2.086176572214125, 2.941076318462707, 3.462428634579362, 4.162159063883735, 4.632217587565054, 5.555358639378547, 6.063364069977190, 6.550535996952960, 7.116979017774687, 7.867724491235969, 8.468746383456718, 8.798999097815738, 9.225262316772368, 10.02774828589883, 10.46530650687727, 10.86066842768618, 11.54732693065998, 12.15715069836205, 12.63366846960862, 13.34969204975736, 13.54489481631051, 14.12745895252577, 14.50546018511346
