| L(s) = 1 | + 2-s − 1.84·3-s + 4-s + 5-s − 1.84·6-s − 0.0938·7-s + 8-s + 0.396·9-s + 10-s − 3.43·11-s − 1.84·12-s − 3.36·13-s − 0.0938·14-s − 1.84·15-s + 16-s + 2.56·17-s + 0.396·18-s + 5.98·19-s + 20-s + 0.173·21-s − 3.43·22-s − 2.45·23-s − 1.84·24-s + 25-s − 3.36·26-s + 4.79·27-s − 0.0938·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.06·3-s + 0.5·4-s + 0.447·5-s − 0.752·6-s − 0.0354·7-s + 0.353·8-s + 0.132·9-s + 0.316·10-s − 1.03·11-s − 0.532·12-s − 0.932·13-s − 0.0250·14-s − 0.475·15-s + 0.250·16-s + 0.621·17-s + 0.0934·18-s + 1.37·19-s + 0.223·20-s + 0.0377·21-s − 0.731·22-s − 0.511·23-s − 0.376·24-s + 0.200·25-s − 0.659·26-s + 0.923·27-s − 0.0177·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 1.84T + 3T^{2} \) |
| 7 | \( 1 + 0.0938T + 7T^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 5.98T + 19T^{2} \) |
| 23 | \( 1 + 2.45T + 23T^{2} \) |
| 29 | \( 1 - 5.77T + 29T^{2} \) |
| 31 | \( 1 + 2.23T + 31T^{2} \) |
| 37 | \( 1 - 0.381T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 3.15T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 7.89T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 0.457T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 4.78T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 8.50T + 79T^{2} \) |
| 83 | \( 1 - 3.35T + 83T^{2} \) |
| 89 | \( 1 + 7.15T + 89T^{2} \) |
| 97 | \( 1 - 6.47T + 97T^{2} \) |
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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
−7.77183462765858463560790662485, −7.25896388409842969682335936256, −6.30927435957676977034978592821, −5.76558243488823538182762666617, −5.02692399316050178148620637389, −4.77555139746401012555719002524, −3.30641418638923445592721100779, −2.67975019184401300911856209571, −1.43217437445407826362081816543, 0,
1.43217437445407826362081816543, 2.67975019184401300911856209571, 3.30641418638923445592721100779, 4.77555139746401012555719002524, 5.02692399316050178148620637389, 5.76558243488823538182762666617, 6.30927435957676977034978592821, 7.25896388409842969682335936256, 7.77183462765858463560790662485
