| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 2·11-s + 12-s + 6·13-s + 16-s + 6·17-s − 18-s − 19-s − 2·22-s − 8·23-s − 24-s − 5·25-s − 6·26-s + 27-s − 2·29-s + 8·31-s − 32-s + 2·33-s − 6·34-s + 36-s + 6·37-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 + 0.603·11-s + 0.288·12-s + 1.66·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.229·19-s − 0.426·22-s − 1.66·23-s − 0.204·24-s − 25-s − 1.17·26-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.348·33-s − 1.02·34-s + 1/6·36-s + 0.986·37-s + 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.166604642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.166604642\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.203849439908142588445471840593, −7.69882153787870957073978234174, −6.64675162962275118982938371464, −6.14049809384263008078071389020, −5.43645382923940338440601871003, −3.96186880992894415939831123312, −3.78115297770820838308775928907, −2.66133136426792886537417337928, −1.69121321868107781517761444841, −0.895430758576028707484351822935,
0.895430758576028707484351822935, 1.69121321868107781517761444841, 2.66133136426792886537417337928, 3.78115297770820838308775928907, 3.96186880992894415939831123312, 5.43645382923940338440601871003, 6.14049809384263008078071389020, 6.64675162962275118982938371464, 7.69882153787870957073978234174, 8.203849439908142588445471840593
