L(s) = 1 | + 2-s − 3-s + 4-s + 0.554·5-s − 6-s − 7-s + 8-s + 9-s + 0.554·10-s − 0.356·11-s − 12-s − 14-s − 0.554·15-s + 16-s − 3.80·17-s + 18-s + 5.74·19-s + 0.554·20-s + 21-s − 0.356·22-s − 2.44·23-s − 24-s − 4.69·25-s − 27-s − 28-s + 0.405·29-s − 0.554·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.248·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.175·10-s − 0.107·11-s − 0.288·12-s − 0.267·14-s − 0.143·15-s + 0.250·16-s − 0.922·17-s + 0.235·18-s + 1.31·19-s + 0.124·20-s + 0.218·21-s − 0.0760·22-s − 0.509·23-s − 0.204·24-s − 0.938·25-s − 0.192·27-s − 0.188·28-s + 0.0753·29-s − 0.101·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 0.554T + 5T^{2} \) |
| 11 | \( 1 + 0.356T + 11T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 - 5.74T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 0.405T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 - 3.54T + 37T^{2} \) |
| 41 | \( 1 + 3.26T + 41T^{2} \) |
| 43 | \( 1 + 6.04T + 43T^{2} \) |
| 47 | \( 1 - 0.603T + 47T^{2} \) |
| 53 | \( 1 + 1.35T + 53T^{2} \) |
| 59 | \( 1 - 5.54T + 59T^{2} \) |
| 61 | \( 1 - 4.38T + 61T^{2} \) |
| 67 | \( 1 + 15.2T + 67T^{2} \) |
| 71 | \( 1 - 7.42T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 1.76T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 4.80T + 89T^{2} \) |
| 97 | \( 1 - 2.65T + 97T^{2} \) |
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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
−7.34012185087594837395895270292, −6.78853367155120123597976858349, −6.00934759650516386315192299300, −5.55184323085577536378974258414, −4.80294716237811291468055938445, −4.03309464453287665316344101915, −3.28189637712399890644047787118, −2.33795864867247457927094410700, −1.41401729874899893507491471304, 0,
1.41401729874899893507491471304, 2.33795864867247457927094410700, 3.28189637712399890644047787118, 4.03309464453287665316344101915, 4.80294716237811291468055938445, 5.55184323085577536378974258414, 6.00934759650516386315192299300, 6.78853367155120123597976858349, 7.34012185087594837395895270292