| L(s) = 1 | − 1.21·2-s − 1.31·3-s − 0.525·4-s + 1.59·6-s + 2.90·7-s + 3.06·8-s − 1.28·9-s − 0.214·11-s + 0.688·12-s − 3.52·14-s − 2.67·16-s − 6.42·17-s + 1.55·18-s − 2.21·19-s − 3.80·21-s + 0.260·22-s − 4.68·23-s − 4.02·24-s + 5.61·27-s − 1.52·28-s + 8.70·29-s + 5.59·31-s − 2.88·32-s + 0.280·33-s + 7.80·34-s + 0.673·36-s + 2.28·37-s + ⋯ |
| L(s) = 1 | − 0.858·2-s − 0.756·3-s − 0.262·4-s + 0.649·6-s + 1.09·7-s + 1.08·8-s − 0.426·9-s − 0.0646·11-s + 0.198·12-s − 0.942·14-s − 0.668·16-s − 1.55·17-s + 0.366·18-s − 0.507·19-s − 0.830·21-s + 0.0554·22-s − 0.977·23-s − 0.820·24-s + 1.08·27-s − 0.288·28-s + 1.61·29-s + 1.00·31-s − 0.510·32-s + 0.0489·33-s + 1.33·34-s + 0.112·36-s + 0.374·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5826878486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5826878486\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 3 | \( 1 + 1.31T + 3T^{2} \) |
| 7 | \( 1 - 2.90T + 7T^{2} \) |
| 11 | \( 1 + 0.214T + 11T^{2} \) |
| 17 | \( 1 + 6.42T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 + 4.68T + 23T^{2} \) |
| 29 | \( 1 - 8.70T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 2.28T + 37T^{2} \) |
| 41 | \( 1 + 3.05T + 41T^{2} \) |
| 43 | \( 1 + 6.36T + 43T^{2} \) |
| 47 | \( 1 - 1.09T + 47T^{2} \) |
| 53 | \( 1 - 6.23T + 53T^{2} \) |
| 59 | \( 1 - 9.26T + 59T^{2} \) |
| 61 | \( 1 + 0.280T + 61T^{2} \) |
| 67 | \( 1 + 7.76T + 67T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 9.52T + 83T^{2} \) |
| 89 | \( 1 - 5.61T + 89T^{2} \) |
| 97 | \( 1 + 18.0T + 97T^{2} \) |
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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
−8.446216210367709168964377856719, −8.012746169188180045438240689918, −6.93740163731884933828338196579, −6.32898869675316710047529373754, −5.35729812918154008483842685077, −4.64153768146999880598130890741, −4.21298896773415036912319609113, −2.62404347696558867506874080767, −1.65128635682749552464347414211, −0.52546808418965433793635862643,
0.52546808418965433793635862643, 1.65128635682749552464347414211, 2.62404347696558867506874080767, 4.21298896773415036912319609113, 4.64153768146999880598130890741, 5.35729812918154008483842685077, 6.32898869675316710047529373754, 6.93740163731884933828338196579, 8.012746169188180045438240689918, 8.446216210367709168964377856719
