| L(s) = 1 | − 5i·2-s + 6i·3-s + 7·4-s + 30·6-s + 244i·7-s − 195i·8-s + 207·9-s + 794·11-s + 42i·12-s − 169i·13-s + 1.22e3·14-s − 751·16-s + 1.53e3i·17-s − 1.03e3i·18-s − 2.70e3·19-s + ⋯ |
| L(s) = 1 | − 0.883i·2-s + 0.384i·3-s + 0.218·4-s + 0.340·6-s + 1.88i·7-s − 1.07i·8-s + 0.851·9-s + 1.97·11-s + 0.0841i·12-s − 0.277i·13-s + 1.66·14-s − 0.733·16-s + 1.28i·17-s − 0.752i·18-s − 1.71·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.779432081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.779432081\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 2 | \( 1 + 5iT - 32T^{2} \) |
| 3 | \( 1 - 6iT - 243T^{2} \) |
| 7 | \( 1 - 244iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 794T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.53e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 702iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.03e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.63e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.05e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 294T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.61e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 3.02e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 626iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.80e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.24e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.73e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.46e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.17e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 7.97e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.80e4iT - 8.58e9T^{2} \) |
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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
−10.90704668785456674378082875608, −10.00279361484659106920487348075, −9.136078248987469149855786524773, −8.432270618504920561227273063467, −6.61946179173071080713170516950, −6.14619396535552541230593474087, −4.48012680351800343502577175334, −3.55006360939905682187576633600, −2.21792017983873206676462031995, −1.40034635527023718921485214108,
0.74977133569411520478731274601, 1.84587647708430760271383295688, 3.84284579471849154955712984821, 4.59627163949371334436113529157, 6.36234121000902572909798993348, 6.94583284412465461775232356622, 7.34318604133491691675136190807, 8.586648962569661670380884630007, 9.730524335571496620734177231365, 10.75748950898616657756193283329
