| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 4·7-s − 8-s + 9-s − 2·10-s − 12-s + 4·13-s + 4·14-s − 2·15-s + 16-s − 18-s + 8·19-s + 2·20-s + 4·21-s + 24-s − 25-s − 4·26-s − 27-s − 4·28-s + 2·30-s − 10·31-s − 32-s − 8·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.288·12-s + 1.10·13-s + 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s + 1.83·19-s + 0.447·20-s + 0.872·21-s + 0.204·24-s − 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.755·28-s + 0.365·30-s − 1.79·31-s − 0.176·32-s − 1.35·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019278680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019278680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−12.82270087482201, −12.59643218237906, −12.11570117065445, −11.52196106833077, −11.02879664943622, −10.57118455655054, −10.20518278809115, −9.618628919973731, −9.324432703651396, −9.082681464231513, −8.399220728852193, −7.598385165715356, −7.322399588561703, −6.670497877754104, −6.332437002637390, −5.847891113942401, −5.453227662970648, −5.007053347488615, −3.918890417938391, −3.446173004519171, −3.162463572779330, −2.270828115036414, −1.670609156727527, −1.098237989325186, −0.3599834025202036,
0.3599834025202036, 1.098237989325186, 1.670609156727527, 2.270828115036414, 3.162463572779330, 3.446173004519171, 3.918890417938391, 5.007053347488615, 5.453227662970648, 5.847891113942401, 6.332437002637390, 6.670497877754104, 7.322399588561703, 7.598385165715356, 8.399220728852193, 9.082681464231513, 9.324432703651396, 9.618628919973731, 10.20518278809115, 10.57118455655054, 11.02879664943622, 11.52196106833077, 12.11570117065445, 12.59643218237906, 12.82270087482201
