| L(s) = 1 | − 1.95·2-s − 4.19·4-s + 5·5-s + 23.7·8-s − 9.75·10-s − 62.2·11-s − 87.5·13-s − 12.8·16-s + 41.9·17-s + 116.·19-s − 20.9·20-s + 121.·22-s − 25.7·23-s + 25·25-s + 170.·26-s + 277.·29-s − 123.·31-s − 165.·32-s − 81.7·34-s + 127.·37-s − 227.·38-s + 118.·40-s + 336.·41-s − 124.·43-s + 260.·44-s + 50.1·46-s + 227.·47-s + ⋯ |
| L(s) = 1 | − 0.689·2-s − 0.524·4-s + 0.447·5-s + 1.05·8-s − 0.308·10-s − 1.70·11-s − 1.86·13-s − 0.200·16-s + 0.597·17-s + 1.40·19-s − 0.234·20-s + 1.17·22-s − 0.233·23-s + 0.200·25-s + 1.28·26-s + 1.77·29-s − 0.712·31-s − 0.912·32-s − 0.412·34-s + 0.564·37-s − 0.970·38-s + 0.470·40-s + 1.28·41-s − 0.440·43-s + 0.894·44-s + 0.160·46-s + 0.706·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.95T + 8T^{2} \) |
| 11 | \( 1 + 62.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 41.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 116.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 25.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 277.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 123.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 127.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 336.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 124.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 227.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 200.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 380.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 466.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 289.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 60.3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 796.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 676.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 966.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 496.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−8.213530645538839329379572769969, −7.65201216315182887398586254762, −7.17881359541735303159227654811, −5.73509437841595655732187738670, −5.10741010306437621465060666643, −4.54068638812264590900811864706, −3.04863819020675889945559844247, −2.31675897918283451286489984417, −0.977002217401778721518699268114, 0,
0.977002217401778721518699268114, 2.31675897918283451286489984417, 3.04863819020675889945559844247, 4.54068638812264590900811864706, 5.10741010306437621465060666643, 5.73509437841595655732187738670, 7.17881359541735303159227654811, 7.65201216315182887398586254762, 8.213530645538839329379572769969
