| L(s) = 1 | − 3-s + 5-s + 2.89·7-s + 9-s + 3.43·11-s − 4.72·13-s − 15-s + 2.61·17-s + 2.28·19-s − 2.89·21-s − 23-s + 25-s − 27-s + 2.53·29-s + 6.19·31-s − 3.43·33-s + 2.89·35-s + 4.89·37-s + 4.72·39-s + 10.4·41-s − 9.01·43-s + 45-s − 2.28·47-s + 1.40·49-s − 2.61·51-s + 0.682·53-s + 3.43·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.09·7-s + 0.333·9-s + 1.03·11-s − 1.31·13-s − 0.258·15-s + 0.634·17-s + 0.523·19-s − 0.632·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.470·29-s + 1.11·31-s − 0.597·33-s + 0.489·35-s + 0.805·37-s + 0.757·39-s + 1.63·41-s − 1.37·43-s + 0.149·45-s − 0.332·47-s + 0.200·49-s − 0.366·51-s + 0.0936·53-s + 0.462·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.012931841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.012931841\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 2.89T + 7T^{2} \) |
| 11 | \( 1 - 3.43T + 11T^{2} \) |
| 13 | \( 1 + 4.72T + 13T^{2} \) |
| 17 | \( 1 - 2.61T + 17T^{2} \) |
| 19 | \( 1 - 2.28T + 19T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 - 6.19T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 9.01T + 43T^{2} \) |
| 47 | \( 1 + 2.28T + 47T^{2} \) |
| 53 | \( 1 - 0.682T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 - 1.74T + 67T^{2} \) |
| 71 | \( 1 + 6.32T + 71T^{2} \) |
| 73 | \( 1 + 1.51T + 73T^{2} \) |
| 79 | \( 1 + 3.86T + 79T^{2} \) |
| 83 | \( 1 + 4.61T + 83T^{2} \) |
| 89 | \( 1 + 6.31T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.859046930892189824708657798971, −7.928337040712817878760512878275, −7.34695462557412478540067917179, −6.44240510095699531839729205502, −5.73325866643328430409506833113, −4.83089412580836331761703644604, −4.41254619764160227226612173778, −3.05700712774328370615736923813, −1.91176098713783698150271267210, −0.970726202147429105428269632510,
0.970726202147429105428269632510, 1.91176098713783698150271267210, 3.05700712774328370615736923813, 4.41254619764160227226612173778, 4.83089412580836331761703644604, 5.73325866643328430409506833113, 6.44240510095699531839729205502, 7.34695462557412478540067917179, 7.928337040712817878760512878275, 8.859046930892189824708657798971
