| L(s) = 1 | + (−2.22 + 0.184i)5-s + (−0.741 + 2.53i)7-s + 2.92i·11-s − 0.587·13-s + 4.81i·17-s − 1.64i·19-s + 2.14·23-s + (4.93 − 0.822i)25-s + 5.05i·29-s − 5.69i·31-s + (1.18 − 5.79i)35-s − 1.78i·37-s − 7.74·41-s − 4.04i·43-s − 0.204i·47-s + ⋯ |
| L(s) = 1 | + (−0.996 + 0.0825i)5-s + (−0.280 + 0.959i)7-s + 0.882i·11-s − 0.163·13-s + 1.16i·17-s − 0.377i·19-s + 0.447·23-s + (0.986 − 0.164i)25-s + 0.938i·29-s − 1.02i·31-s + (0.199 − 0.979i)35-s − 0.293i·37-s − 1.20·41-s − 0.617i·43-s − 0.0297i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2601961266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2601961266\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.22 - 0.184i)T \) |
| 7 | \( 1 + (0.741 - 2.53i)T \) |
| good | 11 | \( 1 - 2.92iT - 11T^{2} \) |
| 13 | \( 1 + 0.587T + 13T^{2} \) |
| 17 | \( 1 - 4.81iT - 17T^{2} \) |
| 19 | \( 1 + 1.64iT - 19T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 - 5.05iT - 29T^{2} \) |
| 31 | \( 1 + 5.69iT - 31T^{2} \) |
| 37 | \( 1 + 1.78iT - 37T^{2} \) |
| 41 | \( 1 + 7.74T + 41T^{2} \) |
| 43 | \( 1 + 4.04iT - 43T^{2} \) |
| 47 | \( 1 + 0.204iT - 47T^{2} \) |
| 53 | \( 1 + 6.67T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 + 4.35iT - 61T^{2} \) |
| 67 | \( 1 - 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 3.07iT - 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 9.71T + 79T^{2} \) |
| 83 | \( 1 + 8.45iT - 83T^{2} \) |
| 89 | \( 1 + 6.74T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−9.207818771252856922978150898266, −8.573888594017909152816969942607, −7.86200578072743523296420278692, −7.06376723708370213236044402087, −6.37435032669122022153757069968, −5.36784783613932479066871773165, −4.58872073534628655578404488918, −3.69423028868548184057979311018, −2.80540526366570793814288282390, −1.71046358555574877145595905664,
0.095443139336393280206166859324, 1.16672560566790534672532828719, 2.93421087183436301232303612599, 3.51726970513801701171253881537, 4.45845649567424029294263587758, 5.15214144706580195665790993274, 6.35407713850549927732941736762, 7.01335373504856025480140409787, 7.75106185536988959349434354583, 8.321342649568592862436970090599
