| L(s) = 1 | + 2.75·2-s + 5.56·4-s − 5-s + 0.587·7-s + 9.81·8-s − 2.75·10-s + 0.892·11-s − 1.31·13-s + 1.61·14-s + 15.8·16-s + 6.89·17-s + 2.78·19-s − 5.56·20-s + 2.45·22-s + 0.636·23-s + 25-s − 3.61·26-s + 3.27·28-s − 5.55·29-s − 9.03·31-s + 24.0·32-s + 18.9·34-s − 0.587·35-s + 0.632·37-s + 7.66·38-s − 9.81·40-s + 7.36·41-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.78·4-s − 0.447·5-s + 0.222·7-s + 3.47·8-s − 0.869·10-s + 0.268·11-s − 0.364·13-s + 0.432·14-s + 3.96·16-s + 1.67·17-s + 0.639·19-s − 1.24·20-s + 0.523·22-s + 0.132·23-s + 0.200·25-s − 0.709·26-s + 0.618·28-s − 1.03·29-s − 1.62·31-s + 4.24·32-s + 3.25·34-s − 0.0993·35-s + 0.104·37-s + 1.24·38-s − 1.55·40-s + 1.15·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.350818853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.350818853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 2.75T + 2T^{2} \) |
| 7 | \( 1 - 0.587T + 7T^{2} \) |
| 11 | \( 1 - 0.892T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 19 | \( 1 - 2.78T + 19T^{2} \) |
| 23 | \( 1 - 0.636T + 23T^{2} \) |
| 29 | \( 1 + 5.55T + 29T^{2} \) |
| 31 | \( 1 + 9.03T + 31T^{2} \) |
| 37 | \( 1 - 0.632T + 37T^{2} \) |
| 41 | \( 1 - 7.36T + 41T^{2} \) |
| 43 | \( 1 + 7.41T + 43T^{2} \) |
| 47 | \( 1 - 6.16T + 47T^{2} \) |
| 53 | \( 1 + 3.18T + 53T^{2} \) |
| 59 | \( 1 + 5.68T + 59T^{2} \) |
| 61 | \( 1 + 4.98T + 61T^{2} \) |
| 67 | \( 1 - 7.78T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 97 | \( 1 - 5.86T + 97T^{2} \) |
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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
−7.82438969107637365903934714917, −7.62498284069755197868116616093, −6.82015993242991985624632348035, −5.94208521902273935002359649050, −5.31553237274059426741348262246, −4.77996882155707588667059890495, −3.64642158549463268920430725952, −3.48102645695819032440271692892, −2.33994961611655180784545503647, −1.31586027371826108649607455120,
1.31586027371826108649607455120, 2.33994961611655180784545503647, 3.48102645695819032440271692892, 3.64642158549463268920430725952, 4.77996882155707588667059890495, 5.31553237274059426741348262246, 5.94208521902273935002359649050, 6.82015993242991985624632348035, 7.62498284069755197868116616093, 7.82438969107637365903934714917
