| L(s) = 1 | − 0.209·2-s − 3-s − 1.95·4-s + 5-s + 0.209·6-s + 0.488·7-s + 0.827·8-s + 9-s − 0.209·10-s + 1.95·12-s + 3.26·13-s − 0.102·14-s − 15-s + 3.73·16-s − 3.33·17-s − 0.209·18-s − 3.86·19-s − 1.95·20-s − 0.488·21-s − 0.267·23-s − 0.827·24-s + 25-s − 0.683·26-s − 27-s − 0.956·28-s + 6.31·29-s + 0.209·30-s + ⋯ |
| L(s) = 1 | − 0.147·2-s − 0.577·3-s − 0.978·4-s + 0.447·5-s + 0.0853·6-s + 0.184·7-s + 0.292·8-s + 0.333·9-s − 0.0661·10-s + 0.564·12-s + 0.906·13-s − 0.0273·14-s − 0.258·15-s + 0.934·16-s − 0.807·17-s − 0.0492·18-s − 0.887·19-s − 0.437·20-s − 0.106·21-s − 0.0557·23-s − 0.168·24-s + 0.200·25-s − 0.133·26-s − 0.192·27-s − 0.180·28-s + 1.17·29-s + 0.0381·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.066589666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.066589666\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.209T + 2T^{2} \) |
| 7 | \( 1 - 0.488T + 7T^{2} \) |
| 13 | \( 1 - 3.26T + 13T^{2} \) |
| 17 | \( 1 + 3.33T + 17T^{2} \) |
| 19 | \( 1 + 3.86T + 19T^{2} \) |
| 23 | \( 1 + 0.267T + 23T^{2} \) |
| 29 | \( 1 - 6.31T + 29T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 - 7.50T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 + 2.26T + 43T^{2} \) |
| 47 | \( 1 + 5.58T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 + 4.51T + 59T^{2} \) |
| 61 | \( 1 - 12.5T + 61T^{2} \) |
| 67 | \( 1 - 8.97T + 67T^{2} \) |
| 71 | \( 1 - 5.31T + 71T^{2} \) |
| 73 | \( 1 - 9.65T + 73T^{2} \) |
| 79 | \( 1 + 1.50T + 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−9.287516733291672931526424848041, −8.532056583333304588767176113233, −7.913196713405384187558359904635, −6.65642240078440472280950777907, −6.09514385827146944836157886490, −5.12092391517403636756992157022, −4.46183801215430366552261265161, −3.56178949009240480949174450352, −2.03661404088264098252352083377, −0.75994276201441569043298062732,
0.75994276201441569043298062732, 2.03661404088264098252352083377, 3.56178949009240480949174450352, 4.46183801215430366552261265161, 5.12092391517403636756992157022, 6.09514385827146944836157886490, 6.65642240078440472280950777907, 7.913196713405384187558359904635, 8.532056583333304588767176113233, 9.287516733291672931526424848041
