| L(s) = 1 | + 2-s − 3.03·3-s + 4-s + 5-s − 3.03·6-s − 3.44·7-s + 8-s + 6.20·9-s + 10-s − 11-s − 3.03·12-s + 3.01·13-s − 3.44·14-s − 3.03·15-s + 16-s + 8.05·17-s + 6.20·18-s − 1.97·19-s + 20-s + 10.4·21-s − 22-s − 1.01·23-s − 3.03·24-s + 25-s + 3.01·26-s − 9.73·27-s − 3.44·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.75·3-s + 0.5·4-s + 0.447·5-s − 1.23·6-s − 1.30·7-s + 0.353·8-s + 2.06·9-s + 0.316·10-s − 0.301·11-s − 0.875·12-s + 0.835·13-s − 0.920·14-s − 0.783·15-s + 0.250·16-s + 1.95·17-s + 1.46·18-s − 0.452·19-s + 0.223·20-s + 2.28·21-s − 0.213·22-s − 0.210·23-s − 0.619·24-s + 0.200·25-s + 0.590·26-s − 1.87·27-s − 0.651·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.456004786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.456004786\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 3.03T + 3T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 13 | \( 1 - 3.01T + 13T^{2} \) |
| 17 | \( 1 - 8.05T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 23 | \( 1 + 1.01T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 - 0.814T + 37T^{2} \) |
| 41 | \( 1 + 8.79T + 41T^{2} \) |
| 47 | \( 1 + 2.28T + 47T^{2} \) |
| 53 | \( 1 + 4.19T + 53T^{2} \) |
| 59 | \( 1 + 0.747T + 59T^{2} \) |
| 61 | \( 1 + 0.594T + 61T^{2} \) |
| 67 | \( 1 - 3.95T + 67T^{2} \) |
| 71 | \( 1 + 5.26T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 - 7.02T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.025868483611539595934253003268, −7.19881987958896822667279844951, −6.40739640591395240641973246683, −6.08995229048178957482556544140, −5.49802313970489275657061197200, −4.88504556839725964808327086592, −3.78182466053203251490992746009, −3.20775187487305503211646815079, −1.76489806522869220371695722771, −0.66360615546239348215409642661,
0.66360615546239348215409642661, 1.76489806522869220371695722771, 3.20775187487305503211646815079, 3.78182466053203251490992746009, 4.88504556839725964808327086592, 5.49802313970489275657061197200, 6.08995229048178957482556544140, 6.40739640591395240641973246683, 7.19881987958896822667279844951, 8.025868483611539595934253003268
