| L(s) = 1 | − 2·4-s − 5-s − 7-s − 11-s + 4·16-s + 4·17-s − 7·19-s + 2·20-s + 3·23-s − 4·25-s + 2·28-s − 3·29-s + 3·31-s + 35-s − 4·37-s − 10·41-s + 43-s + 2·44-s + 47-s + 49-s + 3·53-s + 55-s + 4·59-s + 8·61-s − 8·64-s + 6·67-s − 8·68-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s − 0.377·7-s − 0.301·11-s + 16-s + 0.970·17-s − 1.60·19-s + 0.447·20-s + 0.625·23-s − 4/5·25-s + 0.377·28-s − 0.557·29-s + 0.538·31-s + 0.169·35-s − 0.657·37-s − 1.56·41-s + 0.152·43-s + 0.301·44-s + 0.145·47-s + 1/7·49-s + 0.412·53-s + 0.134·55-s + 0.520·59-s + 1.02·61-s − 64-s + 0.733·67-s − 0.970·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4545065494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4545065494\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.41996567316052, −13.18892904518180, −12.66624216768315, −12.25517454495481, −11.73513555662170, −11.20133602730749, −10.47866819325798, −10.17006737629641, −9.767000187416453, −9.115084160528981, −8.584298483386837, −8.308893442346799, −7.740726435188834, −7.178009288594693, −6.589599329292634, −5.993260219525598, −5.351671789956420, −5.050738123904501, −4.226764282527771, −3.898443716145182, −3.367356027296409, −2.696435472334330, −1.892550996510175, −1.093292216866536, −0.2342557234945885,
0.2342557234945885, 1.093292216866536, 1.892550996510175, 2.696435472334330, 3.367356027296409, 3.898443716145182, 4.226764282527771, 5.050738123904501, 5.351671789956420, 5.993260219525598, 6.589599329292634, 7.178009288594693, 7.740726435188834, 8.308893442346799, 8.584298483386837, 9.115084160528981, 9.767000187416453, 10.17006737629641, 10.47866819325798, 11.20133602730749, 11.73513555662170, 12.25517454495481, 12.66624216768315, 13.18892904518180, 13.41996567316052
