L(s) = 1 | + 5·5-s − 31.3·7-s + 8.94·11-s − 62·13-s + 46·17-s + 107.·19-s − 192.·23-s + 25·25-s + 90·29-s − 152.·31-s − 156.·35-s − 214·37-s + 10·41-s − 67.0·43-s − 398.·47-s + 637.·49-s + 678·53-s + 44.7·55-s + 411.·59-s + 250·61-s − 310·65-s + 49.1·67-s + 366.·71-s + 522·73-s − 280·77-s + 876.·79-s − 380.·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.69·7-s + 0.245·11-s − 1.32·13-s + 0.656·17-s + 1.29·19-s − 1.74·23-s + 0.200·25-s + 0.576·29-s − 0.880·31-s − 0.755·35-s − 0.950·37-s + 0.0380·41-s − 0.237·43-s − 1.23·47-s + 1.85·49-s + 1.75·53-s + 0.109·55-s + 0.907·59-s + 0.524·61-s − 0.591·65-s + 0.0897·67-s + 0.612·71-s + 0.836·73-s − 0.414·77-s + 1.24·79-s − 0.502·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.325151525\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.325151525\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
good | 7 | \( 1 + 31.3T + 343T^{2} \) |
| 11 | \( 1 - 8.94T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46T + 4.91e3T^{2} \) |
| 19 | \( 1 - 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 90T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 214T + 5.06e4T^{2} \) |
| 41 | \( 1 - 10T + 6.89e4T^{2} \) |
| 43 | \( 1 + 67.0T + 7.95e4T^{2} \) |
| 47 | \( 1 + 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 678T + 1.48e5T^{2} \) |
| 59 | \( 1 - 411.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 250T + 2.26e5T^{2} \) |
| 67 | \( 1 - 49.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 366.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 522T + 3.89e5T^{2} \) |
| 79 | \( 1 - 876.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 380.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 970T + 7.04e5T^{2} \) |
| 97 | \( 1 + 934T + 9.12e5T^{2} \) |
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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
−9.558022948169763498860957028814, −8.417994985471116099384001005169, −7.34620666484450842011991198076, −6.77719653018365708186155486420, −5.85726824639262399200624828530, −5.18700649392820428843871395614, −3.83630600896124833998605354063, −3.08895946567299908889273426666, −2.07325714783754065014939677866, −0.54307365479155893514727022506,
0.54307365479155893514727022506, 2.07325714783754065014939677866, 3.08895946567299908889273426666, 3.83630600896124833998605354063, 5.18700649392820428843871395614, 5.85726824639262399200624828530, 6.77719653018365708186155486420, 7.34620666484450842011991198076, 8.417994985471116099384001005169, 9.558022948169763498860957028814