| L(s) = 1 | − 3-s + 0.720·5-s − 1.67·7-s + 9-s + 5.90·11-s − 2.29·13-s − 0.720·15-s − 2.89·17-s + 1.44·19-s + 1.67·21-s − 2.01·23-s − 4.48·25-s − 27-s − 0.177·29-s + 5.80·31-s − 5.90·33-s − 1.20·35-s − 6.65·37-s + 2.29·39-s − 3.89·41-s + 8.38·43-s + 0.720·45-s − 11.5·47-s − 4.17·49-s + 2.89·51-s + 0.153·53-s + 4.25·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.322·5-s − 0.634·7-s + 0.333·9-s + 1.77·11-s − 0.635·13-s − 0.185·15-s − 0.703·17-s + 0.331·19-s + 0.366·21-s − 0.420·23-s − 0.896·25-s − 0.192·27-s − 0.0330·29-s + 1.04·31-s − 1.02·33-s − 0.204·35-s − 1.09·37-s + 0.366·39-s − 0.607·41-s + 1.27·43-s + 0.107·45-s − 1.68·47-s − 0.596·49-s + 0.405·51-s + 0.0210·53-s + 0.573·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 5 | \( 1 - 0.720T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 - 5.90T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 1.44T + 19T^{2} \) |
| 23 | \( 1 + 2.01T + 23T^{2} \) |
| 29 | \( 1 + 0.177T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 + 6.65T + 37T^{2} \) |
| 41 | \( 1 + 3.89T + 41T^{2} \) |
| 43 | \( 1 - 8.38T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 0.153T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 - 9.29T + 61T^{2} \) |
| 67 | \( 1 - 3.04T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 - 5.91T + 79T^{2} \) |
| 83 | \( 1 + 8.74T + 83T^{2} \) |
| 89 | \( 1 + 6.19T + 89T^{2} \) |
| 97 | \( 1 + 3.83T + 97T^{2} \) |
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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
−7.53605341006432649312593933339, −6.63178011054740030090205943961, −6.49982954369145773508515090819, −5.66853846754954144512118747222, −4.79293037226766513639584065942, −4.06442129068329854856278296770, −3.33224877368821999960798924433, −2.17738630378487075049094672368, −1.28365388088033242186105661437, 0,
1.28365388088033242186105661437, 2.17738630378487075049094672368, 3.33224877368821999960798924433, 4.06442129068329854856278296770, 4.79293037226766513639584065942, 5.66853846754954144512118747222, 6.49982954369145773508515090819, 6.63178011054740030090205943961, 7.53605341006432649312593933339
