| L(s) = 1 | − 1.70·2-s + 1.86·3-s + 1.89·4-s − 3.17·6-s − 0.547·7-s − 1.51·8-s + 2.47·9-s + 3.52·12-s + 0.930·14-s + 0.686·16-s − 1.96·17-s − 4.21·18-s + 0.184·19-s − 1.02·21-s − 1.20·23-s − 2.82·24-s + 25-s + 2.75·27-s − 1.03·28-s + 0.891·29-s − 1.96·31-s + 0.349·32-s + 3.34·34-s + 4.68·36-s − 0.313·38-s + 1.73·42-s + 1.47·43-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.86·3-s + 1.89·4-s − 3.17·6-s − 0.547·7-s − 1.51·8-s + 2.47·9-s + 3.52·12-s + 0.930·14-s + 0.686·16-s − 1.96·17-s − 4.21·18-s + 0.184·19-s − 1.02·21-s − 1.20·23-s − 2.82·24-s + 25-s + 2.75·27-s − 1.03·28-s + 0.891·29-s − 1.96·31-s + 0.349·32-s + 3.34·34-s + 4.68·36-s − 0.313·38-s + 1.73·42-s + 1.47·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 383 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 383 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6590763667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6590763667\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 383 | \( 1 - T \) |
| good | 2 | \( 1 + 1.70T + T^{2} \) |
| 3 | \( 1 - 1.86T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 0.547T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.96T + T^{2} \) |
| 19 | \( 1 - 0.184T + T^{2} \) |
| 23 | \( 1 + 1.20T + T^{2} \) |
| 29 | \( 1 - 0.891T + T^{2} \) |
| 31 | \( 1 + 1.96T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.47T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.20T + T^{2} \) |
| 71 | \( 1 - 1.47T + T^{2} \) |
| 73 | \( 1 + 1.20T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−11.03743361250847243799954144757, −10.24081118605281119188308871100, −9.311487134364345841836567427221, −8.961327788546262398094819186663, −8.179640265010831963606294192022, −7.31045249764420186208894013941, −6.57058952502180636325587332990, −4.19840151935758864395708552915, −2.82194953247396000230178449300, −1.89063962636347752068405969415,
1.89063962636347752068405969415, 2.82194953247396000230178449300, 4.19840151935758864395708552915, 6.57058952502180636325587332990, 7.31045249764420186208894013941, 8.179640265010831963606294192022, 8.961327788546262398094819186663, 9.311487134364345841836567427221, 10.24081118605281119188308871100, 11.03743361250847243799954144757
