| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s − 8·11-s − 12·12-s + 42·13-s − 14·14-s + 16·16-s + 2·17-s − 18·18-s − 124·19-s − 21·21-s + 16·22-s − 76·23-s + 24·24-s − 84·26-s − 27·27-s + 28·28-s + 254·29-s − 72·31-s − 32·32-s + 24·33-s − 4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.219·11-s − 0.288·12-s + 0.896·13-s − 0.267·14-s + 1/4·16-s + 0.0285·17-s − 0.235·18-s − 1.49·19-s − 0.218·21-s + 0.155·22-s − 0.689·23-s + 0.204·24-s − 0.633·26-s − 0.192·27-s + 0.188·28-s + 1.62·29-s − 0.417·31-s − 0.176·32-s + 0.126·33-s − 0.0201·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 76 T + p^{3} T^{2} \) |
| 29 | \( 1 - 254 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 398 T + p^{3} T^{2} \) |
| 41 | \( 1 - 462 T + p^{3} T^{2} \) |
| 43 | \( 1 + 212 T + p^{3} T^{2} \) |
| 47 | \( 1 - 264 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 772 T + p^{3} T^{2} \) |
| 61 | \( 1 - 30 T + p^{3} T^{2} \) |
| 67 | \( 1 - 764 T + p^{3} T^{2} \) |
| 71 | \( 1 + 236 T + p^{3} T^{2} \) |
| 73 | \( 1 + 418 T + p^{3} T^{2} \) |
| 79 | \( 1 - 552 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 30 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1190 T + p^{3} T^{2} \) |
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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
−8.956270060815629459429220235000, −8.421773648258684462385716933913, −7.53402376363641726565553000046, −6.53451010627299411133540216567, −5.93673134529339520667387126069, −4.80510176101355675168993501632, −3.78400231562588895042239664406, −2.34606715575916844695857443062, −1.24566253904807803130199676178, 0,
1.24566253904807803130199676178, 2.34606715575916844695857443062, 3.78400231562588895042239664406, 4.80510176101355675168993501632, 5.93673134529339520667387126069, 6.53451010627299411133540216567, 7.53402376363641726565553000046, 8.421773648258684462385716933913, 8.956270060815629459429220235000
