| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s − 4·11-s + 2·12-s − 2·13-s + 16-s − 2·17-s − 18-s + 4·22-s − 2·24-s + 2·26-s − 4·27-s − 8·29-s + 8·31-s − 32-s − 8·33-s + 2·34-s + 36-s + 4·37-s − 4·39-s + 10·41-s − 2·43-s − 4·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.577·12-s − 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.852·22-s − 0.408·24-s + 0.392·26-s − 0.769·27-s − 1.48·29-s + 1.43·31-s − 0.176·32-s − 1.39·33-s + 0.342·34-s + 1/6·36-s + 0.657·37-s − 0.640·39-s + 1.56·41-s − 0.304·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 79 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.19447567315749, −12.73038055061251, −12.22339494678995, −11.66272233817953, −11.07984370683020, −10.85279194213216, −10.25112971891986, −9.689222088202019, −9.458574491329263, −9.012045538630877, −8.468701105686548, −8.017874596356335, −7.610439260100388, −7.505083918080875, −6.680691465586842, −6.166897122220487, −5.650244175516392, −5.073961205973162, −4.421548216705893, −4.007300978595691, −3.140680200326524, −2.713720455555592, −2.582425415224387, −1.757428769311059, −1.287779692382685, 0, 0,
1.287779692382685, 1.757428769311059, 2.582425415224387, 2.713720455555592, 3.140680200326524, 4.007300978595691, 4.421548216705893, 5.073961205973162, 5.650244175516392, 6.166897122220487, 6.680691465586842, 7.505083918080875, 7.610439260100388, 8.017874596356335, 8.468701105686548, 9.012045538630877, 9.458574491329263, 9.689222088202019, 10.25112971891986, 10.85279194213216, 11.07984370683020, 11.66272233817953, 12.22339494678995, 12.73038055061251, 13.19447567315749
