| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 12-s − 13-s + 2·14-s + 16-s + 8·17-s + 18-s + 6·19-s + 2·21-s − 6·23-s + 24-s − 26-s + 27-s + 2·28-s + 4·29-s + 32-s + 8·34-s + 36-s + 2·37-s + 6·38-s − 39-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 + 0.288·12-s − 0.277·13-s + 0.534·14-s + 1/4·16-s + 1.94·17-s + 0.235·18-s + 1.37·19-s + 0.436·21-s − 1.25·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + 0.742·29-s + 0.176·32-s + 1.37·34-s + 1/6·36-s + 0.328·37-s + 0.973·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.545483514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.545483514\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.05328914498867, −12.34812490329580, −11.93610791562620, −11.74135484813272, −11.30362350541446, −10.34472383824925, −10.18594292892573, −9.849137673268797, −9.148570207429703, −8.594429552555829, −7.999024089004384, −7.748081281913722, −7.325602917534633, −6.812546115143960, −5.950564267730077, −5.718259758581788, −5.128938395249999, −4.686299346676414, −4.078204518575932, −3.542676600158720, −3.077455181060956, −2.567593995118365, −1.838168178297979, −1.317497144531320, −0.7061654932047908,
0.7061654932047908, 1.317497144531320, 1.838168178297979, 2.567593995118365, 3.077455181060956, 3.542676600158720, 4.078204518575932, 4.686299346676414, 5.128938395249999, 5.718259758581788, 5.950564267730077, 6.812546115143960, 7.325602917534633, 7.748081281913722, 7.999024089004384, 8.594429552555829, 9.148570207429703, 9.849137673268797, 10.18594292892573, 10.34472383824925, 11.30362350541446, 11.74135484813272, 11.93610791562620, 12.34812490329580, 13.05328914498867
