L(s) = 1 | + 0.434i·2-s − 1.75i·3-s + 1.81·4-s + (0.971 + 2.01i)5-s + 0.763·6-s − 2.56i·7-s + 1.65i·8-s − 0.0918·9-s + (−0.874 + 0.422i)10-s − 2.96·11-s − 3.18i·12-s − 5.29i·13-s + 1.11·14-s + (3.54 − 1.70i)15-s + 2.90·16-s − 2.85i·17-s + ⋯ |
L(s) = 1 | + 0.307i·2-s − 1.01i·3-s + 0.905·4-s + (0.434 + 0.900i)5-s + 0.311·6-s − 0.967i·7-s + 0.585i·8-s − 0.0306·9-s + (−0.276 + 0.133i)10-s − 0.895·11-s − 0.919i·12-s − 1.46i·13-s + 0.297·14-s + (0.914 − 0.441i)15-s + 0.725·16-s − 0.693i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.107352469\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.107352469\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.971 - 2.01i)T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 0.434iT - 2T^{2} \) |
| 3 | \( 1 + 1.75iT - 3T^{2} \) |
| 7 | \( 1 + 2.56iT - 7T^{2} \) |
| 11 | \( 1 + 2.96T + 11T^{2} \) |
| 13 | \( 1 + 5.29iT - 13T^{2} \) |
| 17 | \( 1 + 2.85iT - 17T^{2} \) |
| 19 | \( 1 + 6.43T + 19T^{2} \) |
| 23 | \( 1 + 7.60iT - 23T^{2} \) |
| 29 | \( 1 - 7.01T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 + 0.899iT - 37T^{2} \) |
| 41 | \( 1 - 9.31T + 41T^{2} \) |
| 43 | \( 1 - 12.1iT - 43T^{2} \) |
| 47 | \( 1 - 3.09iT - 47T^{2} \) |
| 53 | \( 1 + 3.14iT - 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 6.41T + 61T^{2} \) |
| 67 | \( 1 + 4.29iT - 67T^{2} \) |
| 71 | \( 1 + 9.51T + 71T^{2} \) |
| 73 | \( 1 - 7.02iT - 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 16.8iT - 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 8.44iT - 97T^{2} \) |
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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
−9.964634904026863429113088725218, −8.179408224477311550064986275799, −7.86179115449184321533825396571, −6.98453919151955693889100144699, −6.53500604149855583790572764528, −5.80330351930080290222234895390, −4.48226738586826985374088398771, −2.81920680487115193530096930138, −2.44490393520334915934778141914, −0.886249164602748126763064873565,
1.69407746850552223129298990782, 2.48336934748033606447021015714, 3.87158813590354667175734409101, 4.66346382702128860769816877324, 5.63357494851191849400654106044, 6.32145327509696405843376419782, 7.45925994879901923901122923036, 8.663007242690544072316232786733, 9.052952968055967539949486181279, 10.10314238884812496552923248434