L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 5·7-s − 8-s + 9-s + 11-s − 12-s + 7·13-s + 5·14-s + 16-s + 6·17-s − 18-s − 19-s + 5·21-s − 22-s + 24-s − 7·26-s − 27-s − 5·28-s − 3·29-s − 2·31-s − 32-s − 33-s − 6·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 1.94·13-s + 1.33·14-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.229·19-s + 1.09·21-s − 0.213·22-s + 0.204·24-s − 1.37·26-s − 0.192·27-s − 0.944·28-s − 0.557·29-s − 0.359·31-s − 0.176·32-s − 0.174·33-s − 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9894808699\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9894808699\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.07632348373934, −13.18096273767358, −12.93261907848447, −12.68799193620905, −11.83569380499036, −11.54012060981074, −10.87672238146907, −10.50403141467542, −9.906253624892800, −9.531411059669170, −9.114078749000986, −8.528714252265728, −7.840466021483453, −7.405191057047979, −6.618522505160677, −6.290655035703288, −5.950712177376354, −5.483735606944883, −4.431486065007696, −3.726642264169275, −3.331931390532346, −2.853050294220554, −1.726567216495461, −1.112401458848832, −0.4362057006597214,
0.4362057006597214, 1.112401458848832, 1.726567216495461, 2.853050294220554, 3.331931390532346, 3.726642264169275, 4.431486065007696, 5.483735606944883, 5.950712177376354, 6.290655035703288, 6.618522505160677, 7.405191057047979, 7.840466021483453, 8.528714252265728, 9.114078749000986, 9.531411059669170, 9.906253624892800, 10.50403141467542, 10.87672238146907, 11.54012060981074, 11.83569380499036, 12.68799193620905, 12.93261907848447, 13.18096273767358, 14.07632348373934