| L(s) = 1 | − 3.14·3-s + 6.89·9-s + 6.61·11-s − 7.89·17-s + 2.51·19-s − 12.2·27-s − 20.7·33-s − 12.7·41-s + 8.48·43-s − 7·49-s + 24.8·51-s − 7.89·57-s − 14.1·59-s − 7.88·67-s + 13.6·73-s + 17.8·81-s + 14.1·83-s − 13.8·89-s − 10·97-s + 45.6·99-s − 4.70·107-s + 0.797·113-s + ⋯ |
| L(s) = 1 | − 1.81·3-s + 2.29·9-s + 1.99·11-s − 1.91·17-s + 0.575·19-s − 2.36·27-s − 3.62·33-s − 1.99·41-s + 1.29·43-s − 49-s + 3.48·51-s − 1.04·57-s − 1.84·59-s − 0.962·67-s + 1.60·73-s + 1.98·81-s + 1.55·83-s − 1.47·89-s − 1.01·97-s + 4.58·99-s − 0.454·107-s + 0.0750·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 6.61T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.89T + 17T^{2} \) |
| 19 | \( 1 - 2.51T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 7.88T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.27348623577398273355510171771, −6.67554628332011968602789862736, −6.39432182352804448785133378649, −5.66096935531862489548177663763, −4.71891998285243366848637964801, −4.35729590380855652173087865507, −3.47396918718054160660739750244, −1.90500964987917448452634149478, −1.13692280252454354787850636147, 0,
1.13692280252454354787850636147, 1.90500964987917448452634149478, 3.47396918718054160660739750244, 4.35729590380855652173087865507, 4.71891998285243366848637964801, 5.66096935531862489548177663763, 6.39432182352804448785133378649, 6.67554628332011968602789862736, 7.27348623577398273355510171771
