| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 6·11-s − 12-s + 4·13-s − 14-s + 16-s − 17-s + 18-s − 19-s + 21-s − 6·22-s − 24-s + 4·26-s − 27-s − 28-s + 9·29-s − 4·31-s + 32-s + 6·33-s − 34-s + 36-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 − 1.80·11-s − 0.288·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.229·19-s + 0.218·21-s − 1.27·22-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s + 1.04·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.056819950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.056819950\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.79445930594082, −15.49583177014298, −14.71929190173813, −14.07062148006515, −13.43891289914553, −13.03079370994581, −12.70560147040487, −11.99619017332072, −11.37834412617222, −10.79024109813588, −10.41959904010816, −9.935008433118466, −8.923182212018122, −8.396113218687102, −7.643901371221647, −7.146382702417572, −6.275666674857917, −5.970042841740040, −5.270643774818002, −4.685005638894600, −4.034242981967654, −3.136990062344267, −2.626565222104285, −1.650492188506032, −0.5494461367663384,
0.5494461367663384, 1.650492188506032, 2.626565222104285, 3.136990062344267, 4.034242981967654, 4.685005638894600, 5.270643774818002, 5.970042841740040, 6.275666674857917, 7.146382702417572, 7.643901371221647, 8.396113218687102, 8.923182212018122, 9.935008433118466, 10.41959904010816, 10.79024109813588, 11.37834412617222, 11.99619017332072, 12.70560147040487, 13.03079370994581, 13.43891289914553, 14.07062148006515, 14.71929190173813, 15.49583177014298, 15.79445930594082
