| L(s) = 1 | − 0.685·2-s − 1.12·3-s − 1.53·4-s + 0.772·6-s + 7-s + 2.41·8-s − 1.72·9-s + 3.79·11-s + 1.72·12-s + 4.67·13-s − 0.685·14-s + 1.40·16-s + 6.81·17-s + 1.18·18-s + 4.06·19-s − 1.12·21-s − 2.59·22-s + 23-s − 2.72·24-s − 3.20·26-s + 5.33·27-s − 1.53·28-s − 1.63·29-s + 10.8·31-s − 5.79·32-s − 4.27·33-s − 4.67·34-s + ⋯ |
| L(s) = 1 | − 0.484·2-s − 0.650·3-s − 0.765·4-s + 0.315·6-s + 0.377·7-s + 0.855·8-s − 0.576·9-s + 1.14·11-s + 0.498·12-s + 1.29·13-s − 0.183·14-s + 0.350·16-s + 1.65·17-s + 0.279·18-s + 0.931·19-s − 0.246·21-s − 0.553·22-s + 0.208·23-s − 0.556·24-s − 0.627·26-s + 1.02·27-s − 0.289·28-s − 0.302·29-s + 1.94·31-s − 1.02·32-s − 0.744·33-s − 0.801·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.262815740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.262815740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 0.685T + 2T^{2} \) |
| 3 | \( 1 + 1.12T + 3T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 - 4.67T + 13T^{2} \) |
| 17 | \( 1 - 6.81T + 17T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 29 | \( 1 + 1.63T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 41 | \( 1 - 5.91T + 41T^{2} \) |
| 43 | \( 1 + 8.83T + 43T^{2} \) |
| 47 | \( 1 + 1.43T + 47T^{2} \) |
| 53 | \( 1 + 5.16T + 53T^{2} \) |
| 59 | \( 1 - 7.98T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 0.0790T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 2.71T + 73T^{2} \) |
| 79 | \( 1 + 8.69T + 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 - 7.18T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.460220368870197940738764575734, −7.971995248650419827429306575728, −6.99467860438954693619856389721, −6.11590171244820455794202059030, −5.50700561814231693898076909604, −4.80448124026115218162328707782, −3.84328672163378393846562031795, −3.15786782046639983821470073175, −1.37452138089670361698708552390, −0.860878125696332452237876421420,
0.860878125696332452237876421420, 1.37452138089670361698708552390, 3.15786782046639983821470073175, 3.84328672163378393846562031795, 4.80448124026115218162328707782, 5.50700561814231693898076909604, 6.11590171244820455794202059030, 6.99467860438954693619856389721, 7.971995248650419827429306575728, 8.460220368870197940738764575734
