| L(s) = 1 | + 2·2-s + 0.535·3-s + 4·4-s + 1.07·6-s + 8·8-s − 26.7·9-s + 57.6·11-s + 2.14·12-s − 88.9·13-s + 16·16-s − 13.2·17-s − 53.4·18-s − 115.·19-s + 115.·22-s − 192.·23-s + 4.28·24-s − 177.·26-s − 28.7·27-s + 247.·29-s + 229.·31-s + 32·32-s + 30.8·33-s − 26.5·34-s − 106.·36-s + 238.·37-s − 231.·38-s − 47.6·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.103·3-s + 0.5·4-s + 0.0728·6-s + 0.353·8-s − 0.989·9-s + 1.57·11-s + 0.0515·12-s − 1.89·13-s + 0.250·16-s − 0.189·17-s − 0.699·18-s − 1.39·19-s + 1.11·22-s − 1.74·23-s + 0.0364·24-s − 1.34·26-s − 0.205·27-s + 1.58·29-s + 1.32·31-s + 0.176·32-s + 0.162·33-s − 0.133·34-s − 0.494·36-s + 1.05·37-s − 0.986·38-s − 0.195·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.053536430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.053536430\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.535T + 27T^{2} \) |
| 11 | \( 1 - 57.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 88.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 13.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 115.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 247.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 213.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 142.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 88.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.10T + 1.48e5T^{2} \) |
| 59 | \( 1 - 639.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 76.8T + 2.26e5T^{2} \) |
| 67 | \( 1 - 549.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 836.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 956.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 719.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 921.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 296.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.54e3T + 9.12e5T^{2} \) |
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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
−8.423803174962269227045444546833, −7.907113546581161258095115452895, −6.66252255560916307139920147944, −6.41964988230723109916672458703, −5.43588358553991407686770786851, −4.43274298587837736550479099515, −3.99658554149908365732083932679, −2.65816440662302952737385636696, −2.20064216305885774138799728636, −0.66106859750675155923249141670,
0.66106859750675155923249141670, 2.20064216305885774138799728636, 2.65816440662302952737385636696, 3.99658554149908365732083932679, 4.43274298587837736550479099515, 5.43588358553991407686770786851, 6.41964988230723109916672458703, 6.66252255560916307139920147944, 7.907113546581161258095115452895, 8.423803174962269227045444546833
