| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 6·6-s − 23·7-s − 8·8-s + 9·9-s − 30·11-s + 12·12-s − 29·13-s + 46·14-s + 16·16-s − 78·17-s − 18·18-s + 149·19-s − 69·21-s + 60·22-s − 150·23-s − 24·24-s + 58·26-s + 27·27-s − 92·28-s − 234·29-s − 217·31-s − 32·32-s − 90·33-s + 156·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.24·7-s − 0.353·8-s + 1/3·9-s − 0.822·11-s + 0.288·12-s − 0.618·13-s + 0.878·14-s + 1/4·16-s − 1.11·17-s − 0.235·18-s + 1.79·19-s − 0.717·21-s + 0.581·22-s − 1.35·23-s − 0.204·24-s + 0.437·26-s + 0.192·27-s − 0.620·28-s − 1.49·29-s − 1.25·31-s − 0.176·32-s − 0.474·33-s + 0.786·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 23 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 29 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 149 T + p^{3} T^{2} \) |
| 23 | \( 1 + 150 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 + 7 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 156 T + p^{3} T^{2} \) |
| 43 | \( 1 - 433 T + p^{3} T^{2} \) |
| 47 | \( 1 + 30 T + p^{3} T^{2} \) |
| 53 | \( 1 - 552 T + p^{3} T^{2} \) |
| 59 | \( 1 + 270 T + p^{3} T^{2} \) |
| 61 | \( 1 - 275 T + p^{3} T^{2} \) |
| 67 | \( 1 + 803 T + p^{3} T^{2} \) |
| 71 | \( 1 - 660 T + p^{3} T^{2} \) |
| 73 | \( 1 - 646 T + p^{3} T^{2} \) |
| 79 | \( 1 - 992 T + p^{3} T^{2} \) |
| 83 | \( 1 - 846 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1488 T + p^{3} T^{2} \) |
| 97 | \( 1 - 319 T + p^{3} T^{2} \) |
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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
−12.08169848805790112871830849969, −10.75963470033737202952304063252, −9.715580222503468816959463814505, −9.181772491829219893931558861413, −7.79678924652181507745354253462, −6.99818726527971649949716632810, −5.56969440598652377685476096237, −3.59073222805582762328566198991, −2.30066864127416624612183458089, 0,
2.30066864127416624612183458089, 3.59073222805582762328566198991, 5.56969440598652377685476096237, 6.99818726527971649949716632810, 7.79678924652181507745354253462, 9.181772491829219893931558861413, 9.715580222503468816959463814505, 10.75963470033737202952304063252, 12.08169848805790112871830849969
