| L(s) = 1 | + 3-s + 5-s + 3·7-s + 9-s + 3·11-s + 2·13-s + 15-s − 19-s + 3·21-s + 23-s + 25-s + 27-s + 4·29-s + 3·33-s + 3·35-s − 4·37-s + 2·39-s + 12·41-s + 11·43-s + 45-s − 6·47-s + 2·49-s − 11·53-s + 3·55-s − 57-s − 6·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.554·13-s + 0.258·15-s − 0.229·19-s + 0.654·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 0.522·33-s + 0.507·35-s − 0.657·37-s + 0.320·39-s + 1.87·41-s + 1.67·43-s + 0.149·45-s − 0.875·47-s + 2/7·49-s − 1.51·53-s + 0.404·55-s − 0.132·57-s − 0.781·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.709695867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.709695867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.92897017256895, −12.40517359331212, −12.17342972912052, −11.34861628835249, −11.04484928428462, −10.79905874169075, −10.04072288065056, −9.571641059858917, −9.129179685622938, −8.742939515551161, −8.250848344931767, −7.761959247205498, −7.420983798673990, −6.496038086076434, −6.453061737976205, −5.749575911511335, −5.007945742084847, −4.766201809237577, −4.003438628335519, −3.724298317195569, −2.939276029503944, −2.320975666303954, −1.840552525767091, −1.208872366093939, −0.7483446712087173,
0.7483446712087173, 1.208872366093939, 1.840552525767091, 2.320975666303954, 2.939276029503944, 3.724298317195569, 4.003438628335519, 4.766201809237577, 5.007945742084847, 5.749575911511335, 6.453061737976205, 6.496038086076434, 7.420983798673990, 7.761959247205498, 8.250848344931767, 8.742939515551161, 9.129179685622938, 9.571641059858917, 10.04072288065056, 10.79905874169075, 11.04484928428462, 11.34861628835249, 12.17342972912052, 12.40517359331212, 12.92897017256895
