L(s) = 1 | + 2-s + 3-s + 4-s + 1.54·5-s + 6-s − 5.16·7-s + 8-s + 9-s + 1.54·10-s − 5.89·11-s + 12-s + 13-s − 5.16·14-s + 1.54·15-s + 16-s + 7.18·17-s + 18-s − 6.16·19-s + 1.54·20-s − 5.16·21-s − 5.89·22-s + 3.21·23-s + 24-s − 2.60·25-s + 26-s + 27-s − 5.16·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.692·5-s + 0.408·6-s − 1.95·7-s + 0.353·8-s + 0.333·9-s + 0.489·10-s − 1.77·11-s + 0.288·12-s + 0.277·13-s − 1.38·14-s + 0.399·15-s + 0.250·16-s + 1.74·17-s + 0.235·18-s − 1.41·19-s + 0.346·20-s − 1.12·21-s − 1.25·22-s + 0.670·23-s + 0.204·24-s − 0.520·25-s + 0.196·26-s + 0.192·27-s − 0.976·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.332618125\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.332618125\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 + 5.16T + 7T^{2} \) |
| 11 | \( 1 + 5.89T + 11T^{2} \) |
| 17 | \( 1 - 7.18T + 17T^{2} \) |
| 19 | \( 1 + 6.16T + 19T^{2} \) |
| 23 | \( 1 - 3.21T + 23T^{2} \) |
| 29 | \( 1 - 5.32T + 29T^{2} \) |
| 31 | \( 1 - 6.16T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 + 0.460T + 41T^{2} \) |
| 43 | \( 1 + 0.464T + 43T^{2} \) |
| 47 | \( 1 + 2.61T + 47T^{2} \) |
| 53 | \( 1 + 2.66T + 53T^{2} \) |
| 59 | \( 1 - 5.23T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 0.908T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.487T + 83T^{2} \) |
| 89 | \( 1 - 1.49T + 89T^{2} \) |
| 97 | \( 1 + 12.5T + 97T^{2} \) |
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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
−8.000847594812217352034610840892, −6.82214969500251380967703085214, −6.48814705222628220070907656290, −5.70336533064966026359071072990, −5.20448328899376319974125700897, −4.13038400424278276453353028220, −3.33605085397854791372185417320, −2.77192004623231761956388825461, −2.27819681231267296188458153631, −0.75405137002776566803773516498,
0.75405137002776566803773516498, 2.27819681231267296188458153631, 2.77192004623231761956388825461, 3.33605085397854791372185417320, 4.13038400424278276453353028220, 5.20448328899376319974125700897, 5.70336533064966026359071072990, 6.48814705222628220070907656290, 6.82214969500251380967703085214, 8.000847594812217352034610840892