| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 5·7-s + 8-s + 9-s − 3·11-s + 12-s − 4·13-s − 5·14-s + 16-s − 6·17-s + 18-s − 5·21-s − 3·22-s + 24-s − 4·26-s + 27-s − 5·28-s + 3·29-s − 31-s + 32-s − 3·33-s − 6·34-s + 36-s − 12·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s − 1.10·13-s − 1.33·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 1.09·21-s − 0.639·22-s + 0.204·24-s − 0.784·26-s + 0.192·27-s − 0.944·28-s + 0.557·29-s − 0.179·31-s + 0.176·32-s − 0.522·33-s − 1.02·34-s + 1/6·36-s − 1.97·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−14.34393405930072, −13.89219276317376, −13.43749888490187, −13.09298545070056, −12.64422383690300, −12.20923066269041, −11.79934828742655, −10.86211705961881, −10.42025710357593, −10.10978449167805, −9.459109387336759, −9.047624067094640, −8.484379149220724, −7.758966979858940, −7.191756332860045, −6.714705015404600, −6.446536596659080, −5.670561144404006, −5.004180771492607, −4.604515401641160, −3.780393655404245, −3.310767209195955, −2.749557567943862, −2.380052472486025, −1.575832435066138, 0, 0,
1.575832435066138, 2.380052472486025, 2.749557567943862, 3.310767209195955, 3.780393655404245, 4.604515401641160, 5.004180771492607, 5.670561144404006, 6.446536596659080, 6.714705015404600, 7.191756332860045, 7.758966979858940, 8.484379149220724, 9.047624067094640, 9.459109387336759, 10.10978449167805, 10.42025710357593, 10.86211705961881, 11.79934828742655, 12.20923066269041, 12.64422383690300, 13.09298545070056, 13.43749888490187, 13.89219276317376, 14.34393405930072
