L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s + 2·11-s + 12-s + 4·13-s − 2·14-s + 16-s + 3·17-s + 18-s − 2·21-s + 2·22-s − 6·23-s + 24-s + 4·26-s + 27-s − 2·28-s − 5·29-s + 31-s + 32-s + 2·33-s + 3·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.436·21-s + 0.426·22-s − 1.25·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.377·28-s − 0.928·29-s + 0.179·31-s + 0.176·32-s + 0.348·33-s + 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.032771830\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.032771830\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.147728419897467686160370184736, −7.62445535535318771300372336150, −6.70581181924992391681724048567, −6.07585340078182618783476685446, −5.54746154479948829458628709310, −4.20632696166754240640102381803, −3.86180585544252780548460023036, −3.06655611841819897046994145376, −2.15090169327151072644179338415, −1.02178556064612628611640235861,
1.02178556064612628611640235861, 2.15090169327151072644179338415, 3.06655611841819897046994145376, 3.86180585544252780548460023036, 4.20632696166754240640102381803, 5.54746154479948829458628709310, 6.07585340078182618783476685446, 6.70581181924992391681724048567, 7.62445535535318771300372336150, 8.147728419897467686160370184736