L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 3·11-s + 12-s + 14-s + 16-s + 3·17-s + 18-s + 5·19-s + 21-s + 3·22-s + 6·23-s + 24-s − 5·25-s + 27-s + 28-s − 3·29-s − 4·31-s + 32-s + 3·33-s + 3·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s + 1.14·19-s + 0.218·21-s + 0.639·22-s + 1.25·23-s + 0.204·24-s − 25-s + 0.192·27-s + 0.188·28-s − 0.557·29-s − 0.718·31-s + 0.176·32-s + 0.522·33-s + 0.514·34-s + 1/6·36-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
\(L(1)\) |
\(\approx\) |
\(5.070156817\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.070156817\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.61033296333440848965148475886, −7.41050604158457718911977471618, −6.54649251029449046549842637035, −5.65816118444786466531912973433, −5.15332984627406569455164982731, −4.20078605597733533031771243492, −3.59633204347625063438465229282, −2.92908234127268278046358107803, −1.88189293402908322204857100573, −1.09105901653179902905242278073,
1.09105901653179902905242278073, 1.88189293402908322204857100573, 2.92908234127268278046358107803, 3.59633204347625063438465229282, 4.20078605597733533031771243492, 5.15332984627406569455164982731, 5.65816118444786466531912973433, 6.54649251029449046549842637035, 7.41050604158457718911977471618, 7.61033296333440848965148475886