L(s) = 1 | − 3-s + 5-s − 3·7-s + 9-s + 2.87·11-s + 4.87·13-s − 15-s − 3.87·17-s − 4.87·19-s + 3·21-s + 23-s + 25-s − 27-s + 1.87·29-s − 3·31-s − 2.87·33-s − 3·35-s + 37-s − 4.87·39-s + 1.87·41-s + 11.7·43-s + 45-s − 0.872·47-s + 2·49-s + 3.87·51-s + 3.87·53-s + 2.87·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.13·7-s + 0.333·9-s + 0.866·11-s + 1.35·13-s − 0.258·15-s − 0.939·17-s − 1.11·19-s + 0.654·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.347·29-s − 0.538·31-s − 0.500·33-s − 0.507·35-s + 0.164·37-s − 0.780·39-s + 0.292·41-s + 1.79·43-s + 0.149·45-s − 0.127·47-s + 0.285·49-s + 0.542·51-s + 0.531·53-s + 0.387·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.477975904\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.477975904\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 2.87T + 11T^{2} \) |
| 13 | \( 1 - 4.87T + 13T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 + 4.87T + 19T^{2} \) |
| 29 | \( 1 - 1.87T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 0.872T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 - 1.87T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 7.87T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.297933909585831816790910274458, −7.14018342556354485140555949463, −6.48731721168165193088041337216, −6.21260844886013901889749400283, −5.49580653896185110172653151757, −4.30352876015498437246279401106, −3.87443312575720363457924189693, −2.82673822831659638447813781400, −1.77986557307551806571387187010, −0.67658412356323317084238697296,
0.67658412356323317084238697296, 1.77986557307551806571387187010, 2.82673822831659638447813781400, 3.87443312575720363457924189693, 4.30352876015498437246279401106, 5.49580653896185110172653151757, 6.21260844886013901889749400283, 6.48731721168165193088041337216, 7.14018342556354485140555949463, 8.297933909585831816790910274458