| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s + 4·13-s + 16-s − 3·17-s − 18-s + 19-s − 22-s − 3·23-s + 24-s − 5·25-s − 4·26-s − 27-s − 9·29-s − 2·31-s − 32-s − 33-s + 3·34-s + 36-s − 7·37-s − 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.229·19-s − 0.213·22-s − 0.625·23-s + 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s − 1.67·29-s − 0.359·31-s − 0.176·32-s − 0.174·33-s + 0.514·34-s + 1/6·36-s − 1.15·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.316794590947454718538747075353, −7.53259325913539386179340129207, −6.87784995348792276292782884110, −5.95475375736342338841751036682, −5.60853815897960330699268784431, −4.24874374197211177370907685875, −3.62923992832075686896391016172, −2.24792901929590427166683754844, −1.32881096542088423713190429622, 0,
1.32881096542088423713190429622, 2.24792901929590427166683754844, 3.62923992832075686896391016172, 4.24874374197211177370907685875, 5.60853815897960330699268784431, 5.95475375736342338841751036682, 6.87784995348792276292782884110, 7.53259325913539386179340129207, 8.316794590947454718538747075353
