| L(s) = 1 | − 2-s − 2.44·3-s + 4-s + 2.44·6-s + 0.449·7-s − 8-s + 2.99·9-s − 4.89·11-s − 2.44·12-s − 4·13-s − 0.449·14-s + 16-s − 4.89·17-s − 2.99·18-s + 8.44·19-s − 1.10·21-s + 4.89·22-s − 0.898·23-s + 2.44·24-s + 4·26-s + 0.449·28-s − 6.44·31-s − 32-s + 11.9·33-s + 4.89·34-s + 2.99·36-s − 37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.41·3-s + 0.5·4-s + 0.999·6-s + 0.169·7-s − 0.353·8-s + 0.999·9-s − 1.47·11-s − 0.707·12-s − 1.10·13-s − 0.120·14-s + 0.250·16-s − 1.18·17-s − 0.707·18-s + 1.93·19-s − 0.240·21-s + 1.04·22-s − 0.187·23-s + 0.499·24-s + 0.784·26-s + 0.0849·28-s − 1.15·31-s − 0.176·32-s + 2.08·33-s + 0.840·34-s + 0.499·36-s − 0.164·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3662497197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3662497197\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 - 0.449T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 8.44T + 19T^{2} \) |
| 23 | \( 1 + 0.898T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 0.449T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 4.89T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 1.55T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 3.79T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.539634505062407237753580038980, −8.382999496451316062056661948183, −7.50247815277804967173666482971, −7.04076746622784735945254544611, −6.00369316019207494121921184293, −5.21832779903434858025545451943, −4.77768532184350806743639605734, −3.12834078045925268178643780899, −1.99164304776830785379259532927, −0.46755908553593451091903809778,
0.46755908553593451091903809778, 1.99164304776830785379259532927, 3.12834078045925268178643780899, 4.77768532184350806743639605734, 5.21832779903434858025545451943, 6.00369316019207494121921184293, 7.04076746622784735945254544611, 7.50247815277804967173666482971, 8.382999496451316062056661948183, 9.539634505062407237753580038980
