L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 12-s − 2·13-s + 14-s + 16-s + 18-s + 8·19-s + 21-s + 24-s − 2·26-s + 27-s + 28-s − 6·29-s + 4·31-s + 32-s + 36-s − 10·37-s + 8·38-s − 2·39-s + 6·41-s + 42-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 + 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s + 1.83·19-s + 0.218·21-s + 0.204·24-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 1/6·36-s − 1.64·37-s + 1.29·38-s − 0.320·39-s + 0.937·41-s + 0.154·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.126097427\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.126097427\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.84149685339839, −12.10068735175593, −11.91568378032180, −11.47226999749552, −10.87601090579392, −10.46018486510765, −9.828312915421131, −9.588650523666196, −8.950086029396467, −8.547348976398977, −7.875574951288122, −7.514827347384450, −7.117989568233809, −6.758189327136887, −5.779250470159955, −5.626746721282268, −5.098188577351564, −4.459541061603764, −4.096024393978286, −3.455375803592424, −2.922752647729670, −2.598064998785853, −1.748417844913970, −1.400334743253242, −0.5372615370725890,
0.5372615370725890, 1.400334743253242, 1.748417844913970, 2.598064998785853, 2.922752647729670, 3.455375803592424, 4.096024393978286, 4.459541061603764, 5.098188577351564, 5.626746721282268, 5.779250470159955, 6.758189327136887, 7.117989568233809, 7.514827347384450, 7.875574951288122, 8.547348976398977, 8.950086029396467, 9.588650523666196, 9.828312915421131, 10.46018486510765, 10.87601090579392, 11.47226999749552, 11.91568378032180, 12.10068735175593, 12.84149685339839