| L(s) = 1 | + 0.470·2-s − 0.577·3-s − 1.77·4-s − 5-s − 0.271·6-s + 7-s − 1.77·8-s − 2.66·9-s − 0.470·10-s + 1.02·12-s − 1.19·13-s + 0.470·14-s + 0.577·15-s + 2.72·16-s + 1.08·17-s − 1.25·18-s + 6.29·19-s + 1.77·20-s − 0.577·21-s − 6.35·23-s + 1.02·24-s + 25-s − 0.563·26-s + 3.27·27-s − 1.77·28-s + 6.79·29-s + 0.271·30-s + ⋯ |
| L(s) = 1 | + 0.332·2-s − 0.333·3-s − 0.889·4-s − 0.447·5-s − 0.110·6-s + 0.377·7-s − 0.628·8-s − 0.888·9-s − 0.148·10-s + 0.296·12-s − 0.332·13-s + 0.125·14-s + 0.149·15-s + 0.680·16-s + 0.263·17-s − 0.295·18-s + 1.44·19-s + 0.397·20-s − 0.126·21-s − 1.32·23-s + 0.209·24-s + 0.200·25-s − 0.110·26-s + 0.630·27-s − 0.336·28-s + 1.26·29-s + 0.0495·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 3 | \( 1 + 0.577T + 3T^{2} \) |
| 13 | \( 1 + 1.19T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 + 6.35T + 23T^{2} \) |
| 29 | \( 1 - 6.79T + 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 + 8.49T + 37T^{2} \) |
| 41 | \( 1 - 9.51T + 41T^{2} \) |
| 43 | \( 1 + 5.07T + 43T^{2} \) |
| 47 | \( 1 - 3.89T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 4.91T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 + 0.0718T + 79T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 + 3.20T + 89T^{2} \) |
| 97 | \( 1 + 9.05T + 97T^{2} \) |
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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
−8.130287825053530702260346931256, −7.46147121916042944857286331639, −6.31084983648432492536861403985, −5.73612648642684013565159991341, −4.91346956072138449106986103680, −4.46897731621169775459569462581, −3.40281573771736381210255959023, −2.76554758369001870681166591513, −1.15056145230138208375398836377, 0,
1.15056145230138208375398836377, 2.76554758369001870681166591513, 3.40281573771736381210255959023, 4.46897731621169775459569462581, 4.91346956072138449106986103680, 5.73612648642684013565159991341, 6.31084983648432492536861403985, 7.46147121916042944857286331639, 8.130287825053530702260346931256
