| L(s) = 1 | − 2-s + 3-s + 4-s − 1.57·5-s − 6-s + 4.87·7-s − 8-s + 9-s + 1.57·10-s + 5.83·11-s + 12-s − 1.66·13-s − 4.87·14-s − 1.57·15-s + 16-s − 1.27·17-s − 18-s − 19-s − 1.57·20-s + 4.87·21-s − 5.83·22-s + 4.60·23-s − 24-s − 2.50·25-s + 1.66·26-s + 27-s + 4.87·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.705·5-s − 0.408·6-s + 1.84·7-s − 0.353·8-s + 0.333·9-s + 0.499·10-s + 1.75·11-s + 0.288·12-s − 0.460·13-s − 1.30·14-s − 0.407·15-s + 0.250·16-s − 0.310·17-s − 0.235·18-s − 0.229·19-s − 0.352·20-s + 1.06·21-s − 1.24·22-s + 0.961·23-s − 0.204·24-s − 0.501·25-s + 0.325·26-s + 0.192·27-s + 0.921·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.310929819\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.310929819\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + 1.57T + 5T^{2} \) |
| 7 | \( 1 - 4.87T + 7T^{2} \) |
| 11 | \( 1 - 5.83T + 11T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 + 1.27T + 17T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 + 4.50T + 29T^{2} \) |
| 31 | \( 1 - 6.49T + 31T^{2} \) |
| 37 | \( 1 + 3.35T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 - 9.86T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 59 | \( 1 - 7.50T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 7.33T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 0.172T + 79T^{2} \) |
| 83 | \( 1 - 9.37T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−8.211403697965463352576906549318, −7.46129665875561999884530313100, −7.06510076807192989788219615707, −6.11854777331465479729017981733, −5.04471757824347988162235390876, −4.28796821653009295314381604185, −3.77201051666557625399377767530, −2.53334282614749085799585107100, −1.69465609139337723278176604355, −0.941824144031553843319637999226,
0.941824144031553843319637999226, 1.69465609139337723278176604355, 2.53334282614749085799585107100, 3.77201051666557625399377767530, 4.28796821653009295314381604185, 5.04471757824347988162235390876, 6.11854777331465479729017981733, 7.06510076807192989788219615707, 7.46129665875561999884530313100, 8.211403697965463352576906549318
