L(s) = 1 | + 1.98·3-s + 1.21·5-s + 4.02·7-s + 0.938·9-s + 2.10·11-s − 1.82·13-s + 2.40·15-s − 2.39·17-s + 19-s + 7.99·21-s + 0.824·23-s − 3.52·25-s − 4.09·27-s + 7.86·29-s + 1.02·31-s + 4.16·33-s + 4.89·35-s − 1.30·37-s − 3.62·39-s + 1.13·41-s + 12.7·43-s + 1.13·45-s − 3.49·47-s + 9.22·49-s − 4.75·51-s − 0.890·53-s + 2.55·55-s + ⋯ |
L(s) = 1 | + 1.14·3-s + 0.543·5-s + 1.52·7-s + 0.312·9-s + 0.633·11-s − 0.506·13-s + 0.622·15-s − 0.581·17-s + 0.229·19-s + 1.74·21-s + 0.171·23-s − 0.705·25-s − 0.787·27-s + 1.45·29-s + 0.184·31-s + 0.725·33-s + 0.826·35-s − 0.214·37-s − 0.580·39-s + 0.176·41-s + 1.95·43-s + 0.169·45-s − 0.509·47-s + 1.31·49-s − 0.666·51-s − 0.122·53-s + 0.343·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.249284109\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.249284109\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 3 | \( 1 - 1.98T + 3T^{2} \) |
| 5 | \( 1 - 1.21T + 5T^{2} \) |
| 7 | \( 1 - 4.02T + 7T^{2} \) |
| 11 | \( 1 - 2.10T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 + 2.39T + 17T^{2} \) |
| 23 | \( 1 - 0.824T + 23T^{2} \) |
| 29 | \( 1 - 7.86T + 29T^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 + 1.30T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 + 3.49T + 47T^{2} \) |
| 53 | \( 1 + 0.890T + 53T^{2} \) |
| 59 | \( 1 - 2.48T + 59T^{2} \) |
| 61 | \( 1 - 6.87T + 61T^{2} \) |
| 67 | \( 1 - 8.07T + 67T^{2} \) |
| 71 | \( 1 - 3.03T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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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
−8.058246146592569397163314558132, −7.66462185431243593587001180650, −6.78466790256812134723447112271, −5.92986244993907337174200699383, −5.06944421820997693236960172733, −4.42662735319040418116332238042, −3.62792389039670244586462338649, −2.51257472099695262473936633702, −2.08826781418735670196745400685, −1.09844156461181472029050653286,
1.09844156461181472029050653286, 2.08826781418735670196745400685, 2.51257472099695262473936633702, 3.62792389039670244586462338649, 4.42662735319040418116332238042, 5.06944421820997693236960172733, 5.92986244993907337174200699383, 6.78466790256812134723447112271, 7.66462185431243593587001180650, 8.058246146592569397163314558132