| L(s) = 1 | + 2.57·2-s − 0.611·3-s + 4.62·4-s + 5-s − 1.57·6-s − 7-s + 6.75·8-s − 2.62·9-s + 2.57·10-s − 2.82·12-s + 4.82·13-s − 2.57·14-s − 0.611·15-s + 8.14·16-s − 1.49·17-s − 6.75·18-s + 2.66·19-s + 4.62·20-s + 0.611·21-s + 2.18·23-s − 4.13·24-s + 25-s + 12.4·26-s + 3.44·27-s − 4.62·28-s + 7.92·29-s − 1.57·30-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.353·3-s + 2.31·4-s + 0.447·5-s − 0.642·6-s − 0.377·7-s + 2.38·8-s − 0.875·9-s + 0.814·10-s − 0.816·12-s + 1.33·13-s − 0.687·14-s − 0.157·15-s + 2.03·16-s − 0.361·17-s − 1.59·18-s + 0.611·19-s + 1.03·20-s + 0.133·21-s + 0.455·23-s − 0.843·24-s + 0.200·25-s + 2.43·26-s + 0.662·27-s − 0.874·28-s + 1.47·29-s − 0.287·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.978983314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.978983314\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.57T + 2T^{2} \) |
| 3 | \( 1 + 0.611T + 3T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 23 | \( 1 - 2.18T + 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 2.91T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 + 7.84T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 + 7.37T + 83T^{2} \) |
| 89 | \( 1 + 8.51T + 89T^{2} \) |
| 97 | \( 1 + 7.78T + 97T^{2} \) |
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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
−8.318544924894529485658363271505, −7.23893779792975626632900607855, −6.47249436833104589793099940612, −6.04123713851935343992687735180, −5.47587606974518306400603609989, −4.72754198882735729813325736928, −3.87311113499332313420918435825, −3.07707964739743253237642831444, −2.46494241322341450304737426759, −1.14033807365131346211276974213,
1.14033807365131346211276974213, 2.46494241322341450304737426759, 3.07707964739743253237642831444, 3.87311113499332313420918435825, 4.72754198882735729813325736928, 5.47587606974518306400603609989, 6.04123713851935343992687735180, 6.47249436833104589793099940612, 7.23893779792975626632900607855, 8.318544924894529485658363271505
