L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 4.47·13-s + 15-s + 4.47·17-s + 21-s − 2.47·23-s + 25-s + 27-s − 0.472·29-s + 2.47·31-s + 35-s − 0.472·37-s + 4.47·39-s − 6.94·41-s + 1.52·43-s + 45-s + 6.47·47-s + 49-s + 4.47·51-s + 2·53-s − 4·59-s + 3.52·61-s + 63-s + 4.47·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 1.24·13-s + 0.258·15-s + 1.08·17-s + 0.218·21-s − 0.515·23-s + 0.200·25-s + 0.192·27-s − 0.0876·29-s + 0.444·31-s + 0.169·35-s − 0.0776·37-s + 0.716·39-s − 1.08·41-s + 0.232·43-s + 0.149·45-s + 0.944·47-s + 0.142·49-s + 0.626·51-s + 0.274·53-s − 0.520·59-s + 0.451·61-s + 0.125·63-s + 0.554·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.013279183\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.013279183\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 + 6.94T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 6.47T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 0.472T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 - 0.944T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−8.664388061843238189325753064565, −7.942008096352279960572492571844, −7.28437247278640288380444857616, −6.26912723110204024189800012017, −5.70349654199735185466378954733, −4.74949927314861669090634041346, −3.81077979732484720244751559696, −3.08688408675868740199599919880, −1.98094352217896926746080943612, −1.09837937907201910720734104826,
1.09837937907201910720734104826, 1.98094352217896926746080943612, 3.08688408675868740199599919880, 3.81077979732484720244751559696, 4.74949927314861669090634041346, 5.70349654199735185466378954733, 6.26912723110204024189800012017, 7.28437247278640288380444857616, 7.942008096352279960572492571844, 8.664388061843238189325753064565