L(s) = 1 | + 2.44·3-s + 7-s + 2.99·9-s + 4.89·11-s + 4.44·13-s + 2·17-s − 1.55·19-s + 2.44·21-s − 2.89·23-s + 6.89·29-s − 8.89·31-s + 11.9·33-s + 2·37-s + 10.8·39-s − 1.10·41-s + 0.898·43-s − 8.89·47-s + 49-s + 4.89·51-s − 10.8·53-s − 3.79·57-s + 1.55·59-s + 3.55·61-s + 2.99·63-s + 8·67-s − 7.10·69-s + 1.10·71-s + ⋯ |
L(s) = 1 | + 1.41·3-s + 0.377·7-s + 0.999·9-s + 1.47·11-s + 1.23·13-s + 0.485·17-s − 0.355·19-s + 0.534·21-s − 0.604·23-s + 1.28·29-s − 1.59·31-s + 2.08·33-s + 0.328·37-s + 1.74·39-s − 0.171·41-s + 0.137·43-s − 1.29·47-s + 0.142·49-s + 0.685·51-s − 1.49·53-s − 0.503·57-s + 0.201·59-s + 0.454·61-s + 0.377·63-s + 0.977·67-s − 0.854·69-s + 0.130·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.697860781\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.697860781\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 23 | \( 1 + 2.89T + 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 8.89T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 1.10T + 41T^{2} \) |
| 43 | \( 1 - 0.898T + 43T^{2} \) |
| 47 | \( 1 + 8.89T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 1.10T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 + 6.89T + 79T^{2} \) |
| 83 | \( 1 - 2.44T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 15.7T + 97T^{2} \) |
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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
−8.597889335380183653525141564849, −8.352832536178618974893166548112, −7.47251406583709945090331298997, −6.58904087027254220018098714846, −5.88780943217056606624382189112, −4.61989811652077315874200043743, −3.75378932186109832527551369097, −3.32012690462247652658375379379, −2.05716228266311701819467963138, −1.28117513753114902008003564599,
1.28117513753114902008003564599, 2.05716228266311701819467963138, 3.32012690462247652658375379379, 3.75378932186109832527551369097, 4.61989811652077315874200043743, 5.88780943217056606624382189112, 6.58904087027254220018098714846, 7.47251406583709945090331298997, 8.352832536178618974893166548112, 8.597889335380183653525141564849