| L(s) = 1 | + 126.·3-s − 9.44e3·7-s − 3.64e3·9-s + 4.87e3·11-s + 1.13e5·13-s + 2.90e5·17-s − 4.23e5·19-s − 1.19e6·21-s + 5.45e5·23-s − 2.95e6·27-s + 4.78e6·29-s + 1.64e6·31-s + 6.17e5·33-s − 1.43e7·37-s + 1.43e7·39-s + 8.04e6·41-s + 4.06e7·43-s + 2.79e7·47-s + 4.89e7·49-s + 3.68e7·51-s − 3.19e7·53-s − 5.36e7·57-s − 5.64e7·59-s − 9.88e7·61-s + 3.44e7·63-s − 2.05e8·67-s + 6.90e7·69-s + ⋯ |
| L(s) = 1 | + 0.902·3-s − 1.48·7-s − 0.185·9-s + 0.100·11-s + 1.10·13-s + 0.844·17-s − 0.745·19-s − 1.34·21-s + 0.406·23-s − 1.06·27-s + 1.25·29-s + 0.319·31-s + 0.0906·33-s − 1.26·37-s + 0.996·39-s + 0.444·41-s + 1.81·43-s + 0.836·47-s + 1.21·49-s + 0.762·51-s − 0.555·53-s − 0.673·57-s − 0.606·59-s − 0.914·61-s + 0.275·63-s − 1.24·67-s + 0.366·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 126.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 9.44e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.87e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.13e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.90e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.23e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.45e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.78e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.64e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.43e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 8.04e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.79e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.64e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.88e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.05e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.27e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.60e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.98e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.08e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.90e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.98e8T + 7.60e17T^{2} \) |
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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
−9.120101837993708915611242997577, −8.620926199944056352166822221226, −7.54764422194548216719685025667, −6.44913814334848777599865157050, −5.76658604115840434289242019396, −4.14675463686919854318966382629, −3.25720212088688041567612945833, −2.66090785157788206413702278010, −1.21528577618610169383571037715, 0,
1.21528577618610169383571037715, 2.66090785157788206413702278010, 3.25720212088688041567612945833, 4.14675463686919854318966382629, 5.76658604115840434289242019396, 6.44913814334848777599865157050, 7.54764422194548216719685025667, 8.620926199944056352166822221226, 9.120101837993708915611242997577
