| L(s) = 1 | + 2-s − 5-s + 7-s − 8-s + 9-s − 10-s + 14-s − 16-s + 18-s + 19-s + 31-s − 35-s + 38-s + 40-s + 41-s − 45-s + 2·47-s − 56-s − 59-s + 62-s + 63-s + 64-s + 2·67-s − 70-s − 71-s − 72-s + 80-s + ⋯ |
| L(s) = 1 | + 2-s − 5-s + 7-s − 8-s + 9-s − 10-s + 14-s − 16-s + 18-s + 19-s + 31-s − 35-s + 38-s + 40-s + 41-s − 45-s + 2·47-s − 56-s − 59-s + 62-s + 63-s + 64-s + 2·67-s − 70-s − 71-s − 72-s + 80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.881000554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.881000554\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.565279294328514056886788649020, −7.77410123715940729153657181110, −7.34288690366067335748202085637, −6.34895692263213641110817612975, −5.43367112117976595783658209696, −4.72102597250644788021082571111, −4.19693135454298735925211135389, −3.57071678691113607232242399600, −2.49215504078017027609476690572, −1.09110013833082705084235024564,
1.09110013833082705084235024564, 2.49215504078017027609476690572, 3.57071678691113607232242399600, 4.19693135454298735925211135389, 4.72102597250644788021082571111, 5.43367112117976595783658209696, 6.34895692263213641110817612975, 7.34288690366067335748202085637, 7.77410123715940729153657181110, 8.565279294328514056886788649020
