| L(s) = 1 | − 1.90·2-s − 1.17·3-s + 2.61·4-s − 5-s + 2.23·6-s − 3.07·8-s + 0.381·9-s + 1.90·10-s − 11-s − 3.07·12-s + 1.17·15-s + 3.23·16-s − 0.726·18-s − 1.61·19-s − 2.61·20-s + 1.90·22-s + 3.61·24-s + 25-s + 0.726·27-s − 0.618·29-s − 2.23·30-s − 1.61·31-s − 3.07·32-s + 1.17·33-s + 0.999·36-s + 3.07·38-s + 3.07·40-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 1.17·3-s + 2.61·4-s − 5-s + 2.23·6-s − 3.07·8-s + 0.381·9-s + 1.90·10-s − 11-s − 3.07·12-s + 1.17·15-s + 3.23·16-s − 0.726·18-s − 1.61·19-s − 2.61·20-s + 1.90·22-s + 3.61·24-s + 25-s + 0.726·27-s − 0.618·29-s − 2.23·30-s − 1.61·31-s − 3.07·32-s + 1.17·33-s + 0.999·36-s + 3.07·38-s + 3.07·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1224096366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1224096366\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 1.90T + T^{2} \) |
| 3 | \( 1 + 1.17T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.61T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 0.618T + T^{2} \) |
| 31 | \( 1 + 1.61T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.90T + T^{2} \) |
| 59 | \( 1 - 0.618T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.90T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.17T + T^{2} \) |
| 79 | \( 1 + 0.618T + T^{2} \) |
| 83 | \( 1 - 1.17T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.035493271066926995697539629042, −8.537141281257181343099954739568, −7.65844980606813388172055313190, −7.24596128657106858647269218093, −6.31132106500299463081141629907, −5.65985165442771193444068549958, −4.48329997288139466548432190592, −3.11546098167201760871228868119, −1.96033816004017012830002554776, −0.44518024874562455784995766073,
0.44518024874562455784995766073, 1.96033816004017012830002554776, 3.11546098167201760871228868119, 4.48329997288139466548432190592, 5.65985165442771193444068549958, 6.31132106500299463081141629907, 7.24596128657106858647269218093, 7.65844980606813388172055313190, 8.537141281257181343099954739568, 9.035493271066926995697539629042
