| L(s) = 1 | + 2.35·2-s − 3-s + 3.55·4-s + 3.69·5-s − 2.35·6-s − 0.801·7-s + 3.66·8-s + 9-s + 8.70·10-s − 2.85·11-s − 3.55·12-s − 1.89·14-s − 3.69·15-s + 1.52·16-s + 2.93·17-s + 2.35·18-s − 2.44·19-s + 13.1·20-s + 0.801·21-s − 6.71·22-s − 7.78·23-s − 3.66·24-s + 8.63·25-s − 27-s − 2.85·28-s + 3.85·29-s − 8.70·30-s + ⋯ |
| L(s) = 1 | + 1.66·2-s − 0.577·3-s + 1.77·4-s + 1.65·5-s − 0.962·6-s − 0.303·7-s + 1.29·8-s + 0.333·9-s + 2.75·10-s − 0.859·11-s − 1.02·12-s − 0.505·14-s − 0.953·15-s + 0.381·16-s + 0.712·17-s + 0.555·18-s − 0.560·19-s + 2.93·20-s + 0.174·21-s − 1.43·22-s − 1.62·23-s − 0.748·24-s + 1.72·25-s − 0.192·27-s − 0.538·28-s + 0.715·29-s − 1.58·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.515463422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.515463422\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.35T + 2T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 + 0.801T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 17 | \( 1 - 2.93T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 + 7.78T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 37 | \( 1 - 7.44T + 37T^{2} \) |
| 41 | \( 1 + 0.850T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 - 2.44T + 47T^{2} \) |
| 53 | \( 1 + 9.96T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 8.12T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 5.40T + 79T^{2} \) |
| 83 | \( 1 - 7.04T + 83T^{2} \) |
| 89 | \( 1 - 1.13T + 89T^{2} \) |
| 97 | \( 1 + 5.94T + 97T^{2} \) |
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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
−11.02428000056674997789312650067, −10.22851001352422597459392482860, −9.541523968179841066512171474452, −7.931350372598345093761783796969, −6.49836467154880290748927830507, −6.08007829497662988317020565119, −5.34355285688175209912160995369, −4.47555183432868305616718971206, −3.00726401005179861311751062407, −1.96556914197670966460621889314,
1.96556914197670966460621889314, 3.00726401005179861311751062407, 4.47555183432868305616718971206, 5.34355285688175209912160995369, 6.08007829497662988317020565119, 6.49836467154880290748927830507, 7.931350372598345093761783796969, 9.541523968179841066512171474452, 10.22851001352422597459392482860, 11.02428000056674997789312650067
